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. 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 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 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 = < Thus only levels with quantum numbers fc differing by a multiple of N can be connected through the poten- For an odd-numbered N-ring a splitting of degenerate As there is no coupling between the ij/+ and the iJ/_, the levels can occur for k = ± N, + 2 N, + 3 N, The usual formula for second order PT for nondegenerate number of levels which have to be compared with the states can be used. For those levels which are not split HMO-model is N, corresponding to the quantum in second order the contribution AE{2) is numbers k = 0, ±1, ±2,..., ±(N-1)/2 in the FE- model. Thus in first order perturbation theory, the (17) degeneracy is not removed from those lowest levels aN n=i \n of the MFE-model which are relevant for the com- mN 2 00 a2 parison with the HMO-model. af(2) __ v " , k, # ——, m e IN, ±|ir«i (2fc)2— (nN)2 For an even-numbered AT-ring, degeneracy can be (18) removed for k= ± 1/2JV, ±2/2N, ±3/2N,.... In the < 1 ® al HMO-model all levels except the lowest and the , (19) 2oc(m N)2 clN2 „ = I m2— n: highest one are doubly degenerate. The lowest levels n =t= m affected by the perturbation K'(0) are thus giving a net splitting of A\^N) in second order PT 5.3. Second Order Perturbation Theory 2 a2 n(mN) ~ , xt\2 ' (22) As the levels with k= ±nN/2 are mixed completely ot(mNY in first order perturbation theory (PT) giving new zero 2 order wave functions ij/±\k\ (11), it seems useful to The additional term oc a in (21), which causes 2) change the whole basis set in the same manner. With the splitting A [ N), is due to the coupling between the wave functions and iA+|o|- This interaction due to the term 1124 G. Taubmann • A Modification of the Free-Electron Model for Cyclic Polyenes [2am cos{mN(j)j] of the potential F'(0) cannot shift easily be seen from (23) that the eigenvalue-equation •A-lmjvi because a level iA-|0| does not exist. of H2 can be block-factorized. For the matrix shown For an odd-numbered ring the lowest pair of levels above, the following three submatrices M0, M+1, and from which degeneracy is removed is iJ/±\nN\. It is M_, exist: worth noting that in first order PT those levels are not splitted linearly in al5 the only contribution in first order is oc a2 • Only a second order splitting cc a\ can M0 H (24) be observed. In conclusion, it has been shown that no pairs of levels relevant for the comparison of HMO- and 2 3 bx b2 b3 V bt b1 b, MFE-model are splitted in second order PT. 2 M J= K 2 b, b2 +1 2 H-t = 25 b2 b1 l b, I' 2' £m > 2 6. Variational Approach b, b, b, 4 bt b2 bl 9 In this section the eigenvalues of the MFE-method The index / of M, gives the quantum number / with the will be calculated to, in principle, arbitrary accuracy. smallest absolute value among the quantum numbers This can be done using a variational approach in the of basis-functions of the submatrix. In this notation of orthogonal basis of the unperturbed FE-model. As an M, the dimension of the matrix Hj to which M, be- important by-product, pairs of levels will be found longs is not indicated explicitly. Inverting the se- from which degeneracy cannot be removed by any quence of the basis functions of one of the two subma- potential of the form (7). trices it can be seen that and Mj have the same The most appropriate basis for setting up the secular-equation. Thus the levels occurring in M_ (0) (0) x matrix H of H = H + where H is the Hamil- and M[ remain degenerate whatever the numerical tonian for the particle in a ring, are the complex func- values of the coefficients of F'(0) (7) are. Increasing tions i(4). For a special matrix we include all the dimension of H^ does not influence the scheme of 0) 4f[ with -(iiJV) < k <(nN). the block-factorization. The new submatrices The matrix H2 of [3]-annulene is given as an in- and Mt still have the same secular-equation. There- structive example. fore degeneracy can be removed only from pairs of 2 levels which belong to M0, viz. with k= ±3, ±6, ±9,..., 6 0 0 b, 0 0 b2 0 0 b3 0 0 b> 0 52 0 0 0 0 b 0 0 0 0 b, 2 b3 ±(3m),..., (meN). 2 0 0 4 0 0 b, 0 0 b2 0 0 b3 0 0 0 32 0 0 0 0 0 0 Now the (2 n N +1) x (2 n N + 1 )-dimensional matrix bt b, b2 b3 ) 0 0 2 0 Hf of the [N] -annulene is considered. It can be seen K 0 2 0 b, 0 0 b2 0 0 0 0 0 0 T2 0 0 0 0 0 0 b, bi b2 from (10) that i/^ ' has non-zero matrix-elements with 2 H2 = K 0 0 b, 0 0 o 0 0 bt 0 0 b2 .(23) 0 0 0 0 0 0 l2 0 0 0 0 the functions i/^J(meN). Thus an arbitrary H^ b 2 b, bt 0 0 b 2 0 0 b j 0 0 22 0 0 b, 0 can be block-factorized into N submatrices. The sub- 0 0 0 0 0 0 32 0 0 ( 0) b3 b 2 b, bt matrix M in which the basis-function t// occurs, is 2 0 0 0 b3 0 0 b 2 0 0 0 0 4 0 0 2 b, 2 (2 n-I-1) -dimensional, the other submatrices are 0 0 b3 0 0 b2 0 0 br 0 0 5 0 2 2 U4 0 0 b3 0 0 b2 0 0 bt 0 0 6 (2n) -dimensional. In a similar way as in the example shown above the submatrices M, are indicated by the The matrix-elements are given in units of a; squares of quantum numbers / with the smallest absolute value negative numbers are indicated as n2 = (— n)2; the off- among its basis functions. Contrary to the notation of diagonal elements are abbreviated as bj = a -/a. Col- 7 the matrix H|;N), no superscript (N) is used to indicate lecting rows and columns which are connected by the number N of C-atoms in the annulene to which the non-zero matrix-elements (e.g. No. 1, 4, 7, 10, and 13 submatrix M, belongs. The quantum numbers occur- with the diagonal elements 62, 32,02, 32, and 62), it can ring in the various submatrices are M o- jNj" 0'-l)N N 0 —N ... -(j-l)N -jN, Mt: (j-l)N + l N +1 1 -JV + 1 ... —(j—\)N + l -jN+1, M_!: jN — 1 (j-l)N-l N-l -1 -N-l ... —(j — l)N—\ , (10) M (j— l)N + 2 N + 2 2 —N + 2 ... —(j—\)N + 2 -jN + 2. G. Taubmann • A Modification of the Free-Electron Model for Cyclic Polyenes 1125 The matrix elements Mjj* of the submatrices M, are ring. The lowest level (k = 0) in this model is non- degenerate, whereas all other levels are twofold degen- M = a _ + kj2 ocS , a = 0, (27) ll jl u 0 erate (k = ± /). The FE-model can be improved by the where kj is the quantum number of the y'-th basis func- introduction of a potential F'() (7) with Af-fold sym- tion of the submatrix. Here, the indices i and j of M/J' metry which takes into account the N C-atoms of an must not be confused with the quantum numbers of [AT]-annulene. Lowering the C^-symmetry of the FE- the basis functions involved. Following the same argu- model to the Cjv-symmetry of the MFE-model re- ments as above, it can be seen that pairs of matrices moves degeneracy from some of the levels of the FE- M; and M_j have the same secular equation. Thus the model. degeneracy of the basis functions of submatrices The effect of V (0) has been investigated using per- which occur as pairs M±/ cannot be removed. turbation theory up to the second order. The only The following values of / are possible: Fourier coefficient in F'() which must not vanish is al. Thus the levels iare certainly splitted in second order PT for every annulene (22); the levels 0+% must Z = 0,±l,±2,...,±^yij, Nodd, (28) 2 be splitted in first order PT only for annulenes with an even number N of C-atoms in the ring (11). / = 0,±l,±2,...,±(y-l^,+y, N even. (29) In a variational approach, the matrix H/; of the Hamiltonian of the MFE-model is set up using the complex basis functions of the particle in a ring up Both [1] E. Hückel, Z. Physik 70, 204 (1931). [5] A. Streitwieser, Molecular Orbital Theory for Organic [2] J. P. Lowe, Quantum Chemistry, Academic Press, San Chemists, John Wiley, New York 1961. Diego 1978. [6] C. A. Coulson, Proc. Roy. Soc. London A 164, 383 [3] E. Heilbronner and H. Bock, Das HMO-Modell und (1938). seine Anwendungen, Verlag Chemie, Weinheim (Bergstr.) [7] K. Razi Naqvi, Ann. Quirn. 86, 337 (1990). 1968. [8] A. A. Frost B. Musulin, J. Chem. Phys. 21, 572 (1953). [4] L. Salem, The Molecular Orbital Theory of Conjugated [9] P. W. Atkins, Molecular Quantum Mechanics, Vol. I, Systems, W. A. Benjamin, Inc., Reading, Massachusetts Clarendon Press, Oxford 1978. 1972. [10] J. R. Piatt, J. Chem. Phys. 17, 484 (1949).