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1121-1126 (1992); Received July 27, 1992 A Modification of the Free-Electron Model for Cyclic Polyenes Gerhard Taubmann Department of Theoretical Chemistry, University of Ulm, Germany Z. Naturforsch. 47 a, 1121-1126 (1992); received July 27, 1992 In the free-electron model (FE) of cyclic polyenes the 7r-electrons are allowed to move freely in a ring. A modification of the FE-model (the MFE-model) is discussed in which the N-fold symmetry due to the N C-atoms of the ring is taken into account by an appropriate potential. The results of the MFE-model are compared with HMO-theory. The sequence of degenerate and non-degenerate levels which determines the chemical properties (aromaticity and anti-aromaticity) of the annulenes is the same for both models. Key words: HMO-theory, Free-electron model, Annulenes, Perturbation theory, Variation theory. 1. Introduction will be compared briefly, and a modification of the FE-model will be suggested. A theory of conjugated 7r-systems must be able to explain Htickel's rule [1] on aromaticity according to which ring systems with (4 n 4- 2) 7t-electrons are stabi- 2. The HMO Model lized (aromatic), whereas ring systems with 4 n 7r-elec- trons are destabilized (anti-aromatic). A cyclic conju- The HMO-theory has been reviewed in various gated polyene with N C-atoms is called [iV]-annulene. articles and books, e.g. [2-5]. The HMO-energies of There are two simple theories of 7r-systems taking into an [AT]-annulene have been given in a closed form by account the 7r-electrons only, the Hückel molecular Coulson [6] orbital (HMO) theory and the free electron (FE) 2 n model. In both models the mutual electrostatic repul- Ej = a 4- 2 ß cos J J = 0,1,..., N — l, (1) sion of the electrons is neglected. N The sequence of degeneracy of the occupied levels in where a is the Coulomb integral and ß is the resonance the ground state of the molecule is the same for both integral. Recently Razi Naqvi [7] has given an alterna- theories. The lowest level is non-degenerate, whereas tive derivation of (1). A simple mnemonic scheme for the other levels up to the HOMO (highest occupied the energy formula has been given by Frost and molecular orbital) are doubly degenerate. If those Musilin [8]. It can be seen from (1) that the lowest and levels are occupied by electrons obeying Pauli's prin- the highest level of [2 iV]-annulenes are nondegenerate, ciple and Hund's rule, each HOMO of the ground whereas for rings with an odd number of C-atoms state of a (4 n 4- 2)-system is occupied by two electrons only the lowest level is nondegenerate. All other levels of opposite spin giving a stable singlett state, whereas are doubly degenerate. The sequence of degeneracy of the HOMOs of the ground state of a 4 «-system are the levels calculated from the HMO-model allows to each occupied by one electron of parallel spin forming predict Hiickel's rule. an unstable diradical. The alternation of aromaticity and anti-aromaticity of the annulenes with increasing number of 7r-elec- 3. The Simple FE-Model trons can be explained by any model which gives the above mentioned scheme of degeneracy. In this paper In the FE-model the 7i-electrons of a cyclic polyene the results of the HMO- and FE-model of annulenes are allowed to move freely in a ring of constant poten- tial, but they cannot leave the ring. The problem is discussed in most books on quantum mechanics [9]; Reprint requests to Dr. G. Taubmann, Department of Theo- retical Chemistry, Albert-Einstein-Allee 11, W-7900 Ulm, here we follow the notation used by Lowe [2]. The Germany. energies and wave functions can be obtained by solv- 0932-0784 / 92 / 1100-1121 $ 01.30/0. - Please order a reprint rather than making your own copy. 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. 1122 G. Taubmann • A Modification of the Free-Electron Model for Cyclic Polyenes ing the Schroder-equation. degeneracy is necessary to explain Hückel's rule. h2 dVW Therefore the influence of a CN-axis on the levels = E\J/{x), 0 < x < L (2) which are degenerate in the simple FE-model has to 2 m dx2 be investigated. The cyclic domain in which the particle can move is A potential with an N-fold axis can be expanded as taken into account by the boundary conditions on the a Fourier-series wave function tj/ and its derivative iJ/'. V' (</>)= £ an cos{nN 4>) + £ bnsm(nN<f)) (6) iA(0) = <ML), <A'(0 ) = iP'(L). (3) Using the angle </> as the independent variable, the Putting the atoms into the appropriate positions, all eigenfunctions can be written in the complex form [2] the sine-terms can be omitted. The cosine-term with n = 0 gives only a constant shift of the energy eigen- 1 <A* = exp (i k (f)), k = 0, ±1, ±2,., 0 G [ 0, 2 7l] . values. The remaining potential is ]/2h (4) The energies are V'(<p)= Z 2ancos(nN(f)) (7) 2 2 h k The factor two is introduced for convenience of the Ev = (5) 2mr further calculation. The coefficient does not vanish, because a Fourier-series beginning from a is not a where m is the mass of the particle and r is the radius 2 model of an N-ring but of a (2N)-ring. No further of the ring. All levels with k 4= 0 are doubly degenerate. restriction on the coefficients a is made. Hence, wave functions with the same absolute value n The energies of the modified FE-model (MFE- | k | can be mixed arbitrarily. The form of ijj given in (4) model) will be calculated using two different approxi- will turn out to be well suited for most of the subse- mation methods. In both cases the simple FE-method quent discussion. Due to the degeneracy of all levels will be used as the unperturbed problem. As the eigen- except the lowest, the same arguments as for the functions of the "particle in a ring" have a very simple HMO model can be applied to explain the alternation structure, the evaluation of the matrix elements is of aromatic and antiaromatic rings with increasing straightforward. A qualitative insight into the struc- (even) number of n electrons. ture of the solutions of the MFE-model can be ob- tained by perturbation theory, which will be worked 4. Modification of the FE-Model out up to the second order. Using direct diagonaliza- tion of the matrix of the Hamiltonian, pairs of levels The simple FE-model outlined in the last section can be found, from which degeneracy cannot be re- makes rather drastic approximations. Assuming that moved by any perturbation of the form of F'(</>) (7). the circumference of the ring is proportional to the number N of C-atoms in it, the radius of the ring is the 5. Perturbation Theory only parameter depending on N which influences the 5.1. Calculation of the Matrix Elements energies. Let us assume that the ring has the form of an ideal circle. Then the particle moving in a ring with The matrix elements V/j a constant potential V = 0 is influenced by a potential with a C^-axis. In a realistic model of a cyclic com- Vi'j = <<Ai IV' I </0 = 2f >Ar(0) V'W ^-(0) d</> (8) pound with N C-atoms placed equidistant on the ring 0 the symmetry of the potential should be CN instead of can be written as a sum of matrix elements which C^. Thus the FE-model can be improved by introduc- are due to the terms [2 an cos (n N (j))] in F'(</>), ing an N-fold symmetry into the potential. Without doing further calculations, Piatt has pointed out that V^ = ^i\2ancos(nN(t>)\il,j>. (9) the introduction of a periodic potential will improve It can be easily seen that the results of the FE-model [10]. Decreasing the symmetry of a problem can remove 0 if \l\±nN i/'<"V > - (10) degeneracy. As pointed out in the former sections, k. (fc + /) 1 |a „ if r l .= +. nN. r G. Taubmann • A Modification of the Free-Electron Model for Cyclic Polyenes 1123 Thus only levels with quantum numbers fc differing by a multiple of N can be connected through the poten- <A+ioi = (12) l/2V tial V'(<f>). 1 iK|k| =—j= cos(fc</>), = — sin(fc</>), ke N yn ]/n 5.2. First Order Perturbation Theory the matrix elements in the new basis are In first order perturbation theory, levels can be <n) shifted only due to matrix-elements connecting levels <<A±|k|l ^' |«A±|m|> = aB(^H-m|.«JV ± <5fc + m,ntf) , (13) of same energy, viz. diagonal matrix-elements or fc, m e N, matrix-elements connecting degenerate levels. As was concluded from (10), there are no diagonal matrix- <<A+lo| l^'(n)l<A+|fc|> = l/2a„ <5fc,„jv, (14) elements Vffi. However, matrix elements connecting öiji Kronecker symbol, degenerate levels are non-zero if k= ±nN/2. ( (*P±\k\ \ I •A+|m|) = 0 » fc, m G N0 .
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