Structure-resonance theory for r-molecular systems based solely on covalent KekulC structures is justified phenomenologically and by reference to recent theoretical work. The idea of antiaromaticity is shown to be a logical extension of the theory, and a concept of a local ring aromaticity or antiaromaticity is quantitatively defined. Estimates of resonance stabilization energies are substantially lower than predictions based on Hiickel MO theory. The resonance theory results agree with those from SCF-LCAO-MO calculations. he recognition of extremely simple algorisms for T counting KekulC structures' and their permutations induced us to test a semiempirical quantum theory with a basis of KekulC structure functions. The mathematical equivalencies3-j between Hiickel molecular orbital (HMO) and valence bond (VB) theories for the benzenoid class of hydrocarbons led us to expect a correspondence of our results to previously known HMO quantities. The results2 were surprising in that it was found that calculated resonance energies, RE, for an extensive series did not correlate well with HMO delocalization energies (correlation coefficient, 0.493 for resonance energy per electron, REPE). Instead there was a congruity with resonance energy values obtained from SCF-LCAO-MO calculations6 (correlation coefficient, REPE, 0.991).The approach that we use is essentially a quantification of the structural resonance theory traditionally applied to structure-reactivity problems in organic chemistry.' In this paper we try to provide some justification for our procedures which will be described in more detail than in the previous communication.Calculations of resonance energies of several classes of aromatic compounds will then be presented, and comparisons with previous MO results and experimental properties will be delineated. We emphasize throughout that our computational procedure is so easily carried out and leads to such sensible results that it should be the method of choice for calculating resonance energies. Applications to the estimation of heats of formation8 and carcinogenic activitiesg of benzenoid hydrocarbons have already given good results, and a description of bond-order relationships will appear in a following article. Resonance TheoryThe formulism of the method is that of VB theory. l 1 For a n-electronic system each energy eigenfunction is written as given in eq 1. The $(u) are structure func-tions, the cf are coefficients, the index u indicates the type of electron distribution associated with the function, and the subscript i numbers the function. After selecting functions, the coefficients and eigenvalues can be evaluated by solving eq 2, where H is the Hamiltonianmatrix and S is the overlap matrix. A conventional approximation is to assume zero overlap of the wave functions so that off-diagonal elements in the matrix involve Hamiltonian integrals only. A significant simplification in obtaining a ground-state eigenvalue is also provided if one assumes a wave function consisting of equal contributions from the c...

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The concepts of normalized irreducible tensorial matrices (NITM) are extended to all finite and compact unitary groups by a development that clarifies their relationship to group theory and matrix algebra. NITM for a unitary group G are shown to be elements of a basis obtained by symmetry adapting to G the matrix basis of a matrix space M ( a , X a*). a z ) is said to be simple. A compound NITM basis of a matrix space results when the space is partitioned into two or more subspaces, each spanned by a simple NITM basis. NITM determined from Griffiths Vcoefficients for the octahedral group are tabulated and used to construct a six-coordinate superposition Hamiltonian.

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Summary An operation loosely described as a type of composition of graphs is studied. Under rather flexible conditions, the resulting composite graphs must be cospectral. This operation is sufficiently powerful to generate eighty‐one cospectral pairs with at most nine vertices. These pairs include the unique smallest cospectral pair, the smallest cospectral connected pair, and one pair of trees with nine vertices. It is felt that this operation provides a unified explanation of cospectrality in several cases that were previously viewed as coincidental.