Magnetic susceptibilities of möbius annulenes

Volume 134, number 4

MAGNETIC SUSCEPTIBILITIES

CHEMICAL PHYSICS LETTERS

6 March 1987

OF Ml)BIUS ANNULENES

Noriyuki MIZOGUCHI Dep~t~ent o~Phy~~cs,Me@’ Coiiege ofP~ur~ucy, Nozuwa, Setagayu-ku, Tokyo 154, Japan Received 2 1October 1986; in final form 19 December 1986

The magnetic susceptibilities of MGbius annulenes are considered within the Ntlckel molecular orbital model. The Hamiltonian for the system in the presence of a magnetic field can be obtained without using any approximations other than those used in London’s method. By using McWeeny’s method the magnetic susc~tibilities of Mobius annulenes are calculated. The results show that Mobius [4n]annulenes have diamagnetic susceptibilities while Mobius [4n+2 Jannulenes have paramagnetic susceptibilities. This indicates that the magnetic susceptibilities of Htlckel and Mobius annulenes show an opposite tendency.

1. Introduction Two types of monocyclic arrays, Hiickel and MSbius annulenes, have been studied in organic chemistry [ 11. The two systems exhibit an entirely opposite structural stability. Htickel systems obey the Htickel which that Hiickel rule states [ 4n + 2 ] annulenes are stable, and Hiickel [ 4~]annulenes are unstable [2,3] *. On the other hand, Mobius systems obey the anti-Hiickel rule which states that Mobius [ 4n Jannulenes are stable, whereas Mobius [ 4n + 21annulenes are unstable ‘. It has also been shown that Htickel [4n + 2]annulenes sustain diamagnetic susceptibility [ 51 whilst Hiickel [ 4~]~ulenes are paramagnetic [ 6,7] +.The recent increasing interest in Hiickel and Mobius annulenes is related to the Hiickel-Mobius concept used in the study of chemical reaction mechanisms [ 81. The magnetic properties of Mobius annulenes have not been investigated. As is the case with their stabilities, it is expected that the magnetic susceptibilities of the two systems

would also show an opposite tendency. The purpose of this paper is to show that this is indeed the case.

2. Hamiltonian matrix for Mbbius annulene in the presence of a magnetic field Following London [ 91, the effects of an applied magnetic field appear only in the Ham~tonian matrix elements for the bonds of a (Htickel) conjugated system, which are given by

h3=ev[i(e~fi)&~l Prs=exp(iKs) Brs.

(1)

Here, e and h are the usual fundamental constants; B is the flux density of the applied field, which is supposed to be uniform and perpendicular to the molecular plane; S, is the signed area of the triangle formed by an arbitrary origin and the bond r-s, & = - &

(2)

and #?=is the field-free resonance integral (3)

* In ref. [ 31 is a generalized and extended Htlckel rule for predicting the stability of an arbitrary conjugated system presented. ’ The anti-Htickel rule for Mobius annulenes can be extended to Mobius polycyclic conjugated systems. + In ref. f 71 it is shown that the sign of the circuit susceptibility for a (Htlckel) altemant polycyclic conjugated system is determined by the size of the circuit.

where xI is the ordinary x atomic orbital in the absence of the field and h is the field-free Hamiltonian in the HMO method. A cyclic polyene with Ncarbon atoms in which each x-orbital is twisted relative to the adjacent x-orbital by an angle A =x/N is called a Mobius [ Nlannulene

0 009-26 14/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Volume 134, number 4

[ 11. The Hamiltonian matrix elements for the bonds are kV, = -8%

for the N-l bond ,

h, =I-%

for the other bonds ,

where j!?$= jx:hxJ

dr cos( x/N) .

(5)

Since the use of twisted n-orbitals has no effect on the derivation of eq. (1)) eqs. (1) and (4) allow us to express the Hamiltonian matrix elements for Mijbius [N] annulenes in the presence of a magnetic field as h,, = - exp (i V,, ) /3$& for the N-l bond , h, =exp(i~J

82

The Hamiltonian Hermitian, h$,=h,

for the other bonds . matrix

given by eq. (6)

bl

(6)

is

because of eq. (2). Eq. (7) guarantees that all the eigenvalues of the Hamiltonian for M6bius [N] annulenes in the presence of the magnetic field are real. Note that eq. (6) involves no approximations other than those used by London [ 91. In McWeeny’s perturbation approach [lo] the London susceptibility of a conjugated system is given by ill1

b2

Fig. 1. Miibius [ 4]annulene (a) and Mijbius [6]annulene with bond alternation (bl, b2): The r-electron energies and magnetic susceptibilities for the graphs bl and b2 are identical.

McWeeny showed that the effects of the magnetic field can be concentrated in just one bond for each ring of a polycyclic conjugated system [ lo,13 1. For annulene, therefore, eq. (8) can be rewritten as follows: x=28(elh)‘~(k,P,,+kSsSn(,,(,~,)s2(C)

(7)

,

a

(4)

,

(10)

where S(C) is the area of the circuit in the system. The choice of the bond appearing on the right-hand side of eq. (10) is arbitrary [ 141 ++. We define a new kind of bond alternation parameter ti&by k =k k ,

(11)

where k, is the usual bond alternation parameter and /?rs denotes the sign of the resonance integral p,s. In this treatment, for example, the bond alternation parameters, l&, for Mijbius [ 41 annulene (the graph shown in fig. la) are kb,,=-l

forthe4-lbond,

R, = 1

for the other bonds ,

(12)

and those for MGbius [ 61annulene (the graph shown in fig. lbl) are

(8) Here, h is the permeability of the vacuum; the sum c (Ts)runs over all the bonds in the system; Prs is the Coulson bond order for bond r--s [ 12 ] and fi (,s)(,u)is the imaginary bond-bond polarizability between bonds r-s and r-u [ lo]. In eq. (8) the resonance integral for bond T-S has been expressed in terms of bond alternation parameter k,, as Brs=kP

,

(9)

where p is the resonance integral for the bond in benzene. Using a suitable unitary transformation, 372

]T6,=-1,

i,2=t?34=&6=k.

(13)

In the graphs shown in fig. 1 we have used the approximation cos(lc/N) z 1 .

(14)

‘+It can be proved analytically that the value of the quantity for bond r-s is the same as that for any other kP,+ k:&,,,,,, bond in the annulene. See also ref. [ 141.

CHEMICAL PHYSICS LETTERS

Volume 134, number 4

This approximation will be used in what follows. The x-electron energy (and the magnetic susceptibility) of Mobius [N] annulene is not dependent on the location of the weight - 1 between two vertices in the graph. For example, the two graphs b 1 and b2 shown in fig. 1 for Mobius [ 6lannulene have the same n-electron energy (and magnetic susceptibility) because a unitary transformation connects the Hamiltonian matrix for the graph bl with that for the graph b2. Since eq. (10) holds regardless of the sign of k,,, we can apply eq. (10) to Mobius annulenes, provided the bond alternation parameters k,, in eq. ( 10) are replaced by the new bond alternation parameters & defined by eq. ( 11) .

3. Magnetic susceptibilities of MiSbiusannulenes We consider a system with a closed-shell configuration. In the absence of bond alternation (in the usual sense), Mobius [4n]annulene has no degenerate non-bonding MOs but Mobius [ 4n + 2 1annulene has [ 1,8]. The former has a closed-shell configuration whereas the latter has not. The introduction of bond alternation produces a splitting of the degenerate non-bonding MOs of Mobius [ 4n + 21annulene [ 15 1, so that Mobius [ 4n + 2 ] annulene with bond alternation has a closed-shell configuration, We calculated from eq. (10) the magnetic susceptibilities of Mobius annulenes. Table 1 shows the values of the magnetic susceptibilities of Mobius [ 4]-, [8]-, [ 12]-, [ 16]-and [20]-annulene without bond ~temation (see fig. 1 and eq. (12)). It can be seen from this table that Miibius [ 4n] annulenes are diaTable 1 Magnetic susceptibilities of Mobius [ 4n] annulenes

X b)

Nz~~’

N=g

N=12

N=16

N=20

0.354 (0.432) c,

0.163 (0.200)

0.107 (0.132)

0.080 (0.096)

0.064 (0.080)

*) The symbol N denotes the number of carbon atoms in the system. w The magnetic susceptibilities X are expressed in units of ~(e/f~)~@( C)‘, S(C) being the area of the circuit. ct The values in parentheses are the topological resonance energy values for Mobius [ 4nfannulenes in units of j?. See ref. [ 191.

6 March 1987

Table 2 Magnetic susceptibilities of Mobius !4n+ 2]annulenes with bond alternation k”’

Nz6b’

N=lO

N=14

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.001 c, -0.006 - 0.020 - 0.050 -0.102 -0.194 -0.363 -0.719 -1.818

- 0.000 - 0.000 -0.001 - 0.005 -0.016 -0.040 - 0.093 -0.214 - 0.604

- 0.000 - 0.000 - 0.000 -0.001 - 0.003 -0.011 - 0.032 -0.089 -0.284

‘) The symbol k is the bond alternation parameter (see tig. 1) . ‘) See footnote a) of table 1. c) See footnote b) of table 1.

magnetic. Table 2 shows the calculated values of the magnetic susceptibilities of Mobius [ 6]-, [ lo] - and [ 14]-annulenes with bond alternation. These systems are supposed to be represented by graphs similar to graph b 1 shown in fig. 1. It can be seen from this table that the Mobius [ 4n+Z]annulenes are paramagnetic regardless of the values of the (usual) bond alternation parameter k. Accordingly it has been demonstrated that the sign of the magnetic susceptibility for Mobius annulenes is determined by the size of the ring only, and not by the value of the bond alternation parameter, and that Mobius [ 4n ] annulenes are diamagnetic while Mobius [ 4n -I-21annulenes are paramagnetic. This result is summarized in table 3, which also shows the stability rules for Htickel and Mobius annulenes and the magnetic susceptibility rule for Htickel annulenes. The magnetic susceptibilities of Htickel and Mobius ann~enes consequently show an entirely opposite tendency. The concept of “aromaticity” is one of the fascinating problems in chemistry [ 161. From a thermoTable 3 Stabilities and magnetic susceptibilities of Htickel and Mobius annulenes Stability

Htlckel Mobius

Magnetic susceptibility

4n+2

4n

4n+2

4n

stable unstable

unstable stable

diama~etic paramaguetic

paramagnetic diamagnetic

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CHEMICAL PHYSICS LETTERS

dynamic point of view stability (or instability) indicates aromaticity (or antiaromaticity ) whilst from a magnetic point of view diamagnetic (or paramagnetic) susceptibility indicates aromaticity (or antiaromaticity ) . In the case of Htlckel annulenes, the two different criteria give consistent predictions [ 17,181 * . It is desirable that this consistency should also hold for Mobius annulenes and our results show that this is indeed the case. It has been demonstrated for Htlckel [ 4n +2]annulenes that the quantity ~/@(e/zi)~&S~( C) is proportional to the Dewar-type resonance energy in HMO theory [ 17,181. It can also be seen from table 1 that there is a definite linear correlation between ~/#?(e/fi)~@~( C) and the topological resonance energy (TRE) for Mobius [ 4nlannulenes. # In ref. [ 181 it is shown that an explicit relationship exists between the topological resonance energy (TRE) and the London magnetic susceptibility for a polycyclic conjugated system, but this relationship does not guarantee the proportionality of the two quantities.

References [ 1] E. Heilbronner, Tetrahedron Letters (1964) 1923. [2] A. Graovac, I. Gutman and N. Trinajstic, Topological approach to the chemistry of conjugated molecules (Springer, Berlin, 1977), and references therein. [ 31 H. Hosoya, K. Hosoi and I. Gutman, Theoret. Chim. Acta 38 (1975) 37. [4] N. Trinajstic, Chemical graph theory (CRC Press, Boca Raton, 1983)) and references therein. [ 51 L. Pauling, J. Chem. Phys. 4 (1936) 673; L. Salem, The molecular-orbital theory of conjugated systems (Benjamin, New York, 1966).

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[ 61 H.C. Longuet-Higgins and L. Salem, Proc. Roy. Sot. A257 (1960) 445; J.A. Pople and K.G. Untch, J. Am. Chem. Sot. 88 (1966) 4811; T. Nakajima and S. Kohda, Bull. Chem. Sot. Japan 39 (1966) 804. [ 71 N. Mizoguchi, Chem. Phys. Letters 106 (1984) 45 1. [S] H.E. Zimmerman, J. Am. Chem. Sot. 88 (1966) 1564; Quantum mechanics for organic chemistry (Academic Press, New York, 1975); M.J.S. Dewar and R.C. Dougherty, The PM0 theory of organic chemistry (Plenum Press, New York, 1975). [9] F. London, J. Phys. Radium 8 (1937) 397. [lo] R. McWeeny, Mol. Phys. 1 (1958) 311. [ 111 A. Veillard, J. Chim. Phys. 59 (1962) 1056. [ 121 C.A. Coulson and H.C. Longuet-Higgins, Proc. Roy. Sot. A191 (1947) 39;Al92 (1947) 16. [ 131 R.B. Mallion, Proc. Roy. Sot. A341 (1975) 429. [ 141 N. Mizoguchi, Bull. Chem. Sot. Japan 56 (1983) 1588. [ 151 Y.N. Chiu, Chem. Phys. Letters 97 (1983) 26; Theoret. Chim. Acta 62 (1983) 403. [ 161 J.A. Elvidge and L.M. Jackman, J. Chem. Sot. (1961) 859; H.J. Dauben Jr., J.D. Wilson and J.L. Laity, J. Am. Chem. Sot. 91 (1969) 1991; F. Sondheimer, Accounts Chem. Res. 5 (1972) 8 1; R.C. Haddon, V.R. Haddon and L.M. Jackman, Topics Current Chem. 16 (1971) 103; I. Agranat and A. Barak, MTP international review of science, organic chemistry, Series 2, Vol. 3. Aromatic compounds, ed. H. Zollinger (Butterworths, London, 1976); J-F. Labarre and F. Crasnier, Topics Current Chem. 24 (1971) 33; G.M. Badger, Aromatic character and aromaticity (Cambridge Univ. Press, Cambridge, 1969). [ 171 R.C. Haddon, J. Am. Chem. Sot. 101 (1979) 1722; J. Aihara, Bull. Chem. Sot. Japan 53 (1980) 1163; B.A. Hess Jr., L.J. Schaad and M. Nakagawa, J. Org. Chem. 42 (1977) 1669. [ 181 J. Aihara, J. Am. Chem. Sot. 103 (1981) 5704. [ 191 P. Ilic, B. Sinkovic and N. Trinajstic, Israel J. Chem. 20 (1980) 258.