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On Alternant Molecules with IdenticalEnergy Spectra:Isospectral Molecules

N. TYUTYULKOV* AND F. DIETZUniversitat Leipzig, Fakultat fur Chemie und Mineralogie, Institut fur Physikalische und Theoretische¨ ¨ ¨ ¨Chemie, D-04103 Leipzig, Germany

G. OLBRICHGmelin Institut fur Anorganische Chemie, D-60444 FrankfurtrMain, Germany¨

Received November 27, 1995; revised manuscript received May 20, 1996; accepted June 25, 1996

ABSTRACT

ŽIt is shown that if a pair of alternant molecules are isospectral they have identical energy.spectra in the topological—Huckel—approximation they are also isospectral, taking into¨

account the electron correlation. The proof is given in the AMO approximation using aHubbard Hamiltonian. Q 1997 John Wiley & Sons, Inc.

Introduction

n the group of the p-conjugated alternantI molecules, there is a subgroup of moleculeswith identical topological spectra—the isospectral

w x Ž w xor cospectral molecules 1]3 see also 4, 5 and.references given therein . The term ‘‘isospectral’’ is

w xtaken from the graph theory 2 . Let us denote byŽ . Ž .e A and e B the Huckel MO energies of a pair¨i i

of alternant molecules A and B with N s 2np-centers.

*Permanent address: Faculty of Chemistry, University ofSofia, BG-1126 Sofia, Bulgaria.

If the molecules are isospectral,

Ž . Ž . Ž .e A s e B , 1i i

i s 1, 2, . . . N. The simplest examples of sucha pair of isospectral molecules are 1,4-divinyl-benzene 1 and 2-phenylbutadiene 2:

1 2

( )International Journal of Quantum Chemistry, Vol. 62, 167]169 1997Q 1997 John Wiley & Sons, Inc. CCC 0020-7608 / 97 / 020167-03

TYUTYULKOV, DIETZ, AND OLBRICH

or the pair 3 and 4:

3 4

Many examples for isospectral systems are givenw xin 1]3 . We want to see how the electron corre-

lation influences the spectrum of isospectralmolecules.

Method

The dependence of the energy spectrum on thetopology and electron correlation was examined inw x w x Ž6, 7 . In framework of the AMO 8, 9 alternant

.molecular orbitals variant of the extendedŽ .Hartree]Fock EHF method using a Hubbard

w x w xHamiltonian 10 , it was shown 6, 7 that theAMO energies E of a homonuclear alternant p-isystem with a singlet ground state are connected

Žwith the HMO energies e by the formula theiAMO energies are independent of the spin vari-

Ž . Ž . .able s g a, b , E a s E b s E :i i i

2 2 2 Ž .E s "6e q d g , 2i i

where b is the resonance integral and g is theŽone-center Coulomb repulsion integral Hubbard

w x.parameter 10 for the carbon p-center. d is thecorrelation correction which satisfies the equation

2n s gr6b 2e2 q d 2g 2Ý ii

2 2 2Ž . Ž .s 1r6 b _ g e q d . 3Ý ii

Since the isospectral molecules have an equalnumber of electrons N s 2n, the following equa-tion holds:

22 2 2Ž . Ž .2n s 1r6b e A q g d AÝ ii

22 2 2Ž . Ž . Ž .s 1r6b e B q g d B . 4Ý ii

Ž . Ž .It follows from 1 and 4 that

Ž . Ž .d A s d B s d

and

Ž . 2 2 2 2 Ž . Ž .E A s 6b e q g d s E B , 5i i i

i s 1, 2, . . . N. In other words, when in the topolog-ical— Huckel— approximation two alternant¨molecules are isospectral, in the EHF—Hubbard—approximation, they have also identicalenergy spectra. The MO energies of the isospectral

Ž .pair 1]2 are in b units

e s 2.2143; e s 1.6751;1 2

e s e s 1.0; e s 0.5392.3 4 5

With a standard value of the ratio brg s 0.4444w x Ž . w7 b s y2.4 eV and g s 5.4 eV , we obtain Eq.Ž .x3 a value of the correlation parameter: d s

Ž .0.1622. The corresponding AMO energies in eVare

E s 5.3896; E s 3.6082;1 2

E s E s 2.5623; E s 1.5748.3 4 5

The considerations in this article are also validw Ž .x Ž .Eq. 5 for alternant systems isospectral trees inwhich the carbon atoms have different valencies,

w xsuch as 11

This approach cannot be extended directly toisospectral systems which are nonalternant or het-eronuclear, because in this case, the relation be-tween the HMO and AMO energies is more com-

w xplicated 12 .

ACKNOWLEDGMENTS

Ž .Two of the authors N. T. and F. D. thank theŽ .Deutsche Forschungsgemeinschaft and F. D. the

Fonds der Chemischen Industrie for financial sup-port.

References

Ž .1. W. C. Herndon, Tetrahedron Lett. 671 1974 .2. W. C. Herndon and M. L. Ellzey, Jr., Tetrahedron 31, 99

Ž .1975 .

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ISOSPECTRAL MOLECULES

3. T. Zivkovic, N. Trinajstic, and M. Randic, Mol. Phys. 30, 517Ž .1975 .

4. A. Graovac, I. Gutman, and N. Trinajstic, Lecture Notes inChemistry, Vol. 4, Topological Approach to the Chemistry

Ž .of Conjugated Molecules Springer, Berlin, 1977 .5. I. Gutman and O. E. Polansky, Mathematical Concepts in

ŽOrganic Chemistry Academic Press, London, Springerw x .Akademie Verlag Berlin, 1986 .

Ž .6. N. Tyutyulkov, Int. J. Quantum Chem. 9, 683 1975 .

7. N. Tyutyulkov, F. Dietz, D. Klein, W. A. Seitz, and T. G.Ž .Schmalz, Int. J. Quantum Chem. 51, 173 1994 .

Ž .8. P.-O. Lowdin, Phys. Rev. 97, 1509 1955 .¨Ž .9. R. Paunz, AMO Method Saunders, Philadelphia, 1967 .

Ž .10. J. Hubbard, Proc. R. Soc. A 276, 238 1963 ; Ibid. 281, 401Ž .1964 .

11. M. Randic, private communication.

Ž .12. N. Tyutyulkov and F. Dietz, Chem. Phys. 171, 293 1993 .

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