Mobile Bond Orders in the Resonance and Molecular Orbital Theories
1.K. Ruedenberg, J. Chem. Phys. 22, 1878 (1954).
2.B. H. Chirgwin and C. A. Coulson, Proc. Roy. Soc. (London) A201, 196 (1950).
3.N. S. Ham and K. Ruedenberg, J. Chem. Phys. 29, 1215 (1958), preceding paper.
4.See reference 1, Eq. (4.89).
5.An alternant hydrocarbon is one in which the N carbon atoms can be divided into two classes (starred and unstarred) in such a way that no two neighbors belong to the same class. Then it can be proved that the eigenvalues are paired, i.e., and the eigenvectors are also paired, i.e., and .
6.In these equations M is a matrix whose rows correspond to starred atoms, whose columns correspond to unstarred atoms, and whose elements always correspond to a starred‐unstarred pair. is the transpose of M.
7.H. C. Longuet‐Higgins, J. Chem. Phys. 18, 265 (1950).
7.M. J. S. Dewar and H. C. Longuet‐Higgins, Proc. Roy. Soc. (London) A214, 482 (1952).
8.Equation (12) holds for even benzenoid alternants with an equal number of starred and unstarred atoms. Longuet‐Higgins(see reference 7) showed that an alternant in which the starred atoms outnumber the unstarred atoms by R, has no KS, det and at least R zero eigenvalues. For nonbenzenoid alternants with an equal number of starred and unstarred atoms, det M (hence det M) can vanish by there being an equal number of positive and negative structures, e.g., cyclobutadiene.
9.L. Pauling, The Nature of the Chemical Bond (Cornell University Press, Ithaca, 1940), p. 142.
10.There are now two matrix equations for MO bond ordersone for the Coulson bond orders which was derived by G. G. Hall, Proc. Roy. Soc. (London) A229, 251 (1955), and the above one for the resonance theory bond orders These equations are and .
11.See reference 1, Eq. (4.92).
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