J . A m . Chem. Soc. 1982, 104, 3541-3542 equally satisfactory results in the absence of trifluoroacetic acid, in fact, but we feel that trifluoroacetic acid may, in some cases, be advantageous as a solvent for high molecular weight permethylated polysaccharides. Analysis of all reactions after workup by G C / M S demonstrated in each case that reductive cleavage was virtually quantitative. In each reaction the product of reductive cleavage was identical with the authentic anhydroalditol derivative as judged by electron impact mass spectrometry, chemical ionization mass spectrometry, ' H N M R spectroscopy, gas-liquid chromatography, and optical rotation. We conclude from these results that reductive cleavage of glycosides is potentially an attractive method for polysaccharide structure determination. Moreover, the reaction conditions described herein are synthetically useful as a means to prepare anhydroalditols, which have proven to be very useful analogues for the study of the mechanisms of carbohydrate-requiring enzymes. Acknowledgment. This investigation was supported by Grant Number CA 15325, awarded by the National Cancer Institute, DHEW.
On the Concept of Graph-Theoretical Individual Ring Resonance Energies William C. Herndon
Department of Chemistry University of Texas at El Paso, El Paso. Texas 79968 Received December 21, 1981 Hiickel molecular orbital theory has been reexamined and redefined in terms of graph-theoretical concepts.'s2 Graph-theoretical definitions of resonance energies (GTRE) have arisen out of this work,3" and the applications of these definitions comprise a sizable part of the chemical graph-theoretical literature. Recent papers have stressed the point that GTRE's can be divided among the individual rings of polycyclic a molecular graphs. Both individual ring a r o m a t i c i e ~ and ~ ~ ~the J ~theory of London diamagnetism",'2 have been interpreted on this basis. The purpose of this communication is to point out that the principal d e f i n i t i ~ n ' - ' ~for , ~ individual ~ ring aromaticies is based on polynomial equations that have imginary roots in several key, nontrivial cases. The roots of these polynomials must be taken to correspond to energy levels, and the existence of imaginary roots therefore obviates the use of these GTRE's in discussing ring aromaticities or susceptibilities due to individual ring currents. This failing of the GTRE definition, when added to other, less formal types of diffic~lties,'~-'~ should lead to caution in the use of the GTRE concept. The details of a GTRE calculation are as follows. The coef( I ) Gutman, I . ; TrinajstiE, N. Top. Curr. Chem. 1973, 42, 49-93. (2) Graovac, A,; Gutman, I.; TrinajstiE, N. Lect. Notes Chem. 1977, 4, 1-123. (3) Gutman, I.; Milun, M.; TrinajstiE, N . MATCH 1975, 1, 171-175. (4) Aihara, J. J . A m . Chem. SOC.1976, 98, 2750-2758. (5) Aihara, J. J . A m . Chem. SOC.1977, 99, 2048-2053. (6) Gutman, I.; Milun, M.; TrinajstiE, N. J . A m . Chem. SOC.1977, 99, 1692-1 704. (7) Gutman, I . ; Bosanac, S. Tetrahedron 1977, 33, 1809-1812. ( 8 ) Bosanac, S.; Gutman, I . Z. Naturforsch. 1977, 32a, 10-12. (9) Gutman, I . J . Chem. SOC.Faraday Trans. 2, 1979, 75, 799-805. ( I O ) Gutman, I . Croat. Chem. Acta 1980, 53, 581-586. ( 1 1 ) Ihara, J. J . A m . Chem. SOC.1979, 101, 5913-5917. (12) Aihara, J. J . A m . Chem. Soc. 1981, 103, 5704-5706. (13) Gutman, I . Chem. Phys. Lett. 1979, 66, 595-597. (14) Gutman, I.; Mohar, B. Chem. Phys. Lett. 1980, 69, 375-377. (15) Gutman, I. Theor. Chim. Acta 1980, 56, 89-92. (16) Gutman, E.; Mohar, B. Chem. Phys. Lett. 1981, 77, 567-570. (17) Aihara, J . Chem. Phys. Lett. 1980, 73, 404-406. This paper replies to criticisms in ref 13. (18) Herndon, W. C. J . Org. Chem. 1981, 46, 2119-2125. (19) E. Heilbronner, submitted for publication. A copy of this article was kindly supplied by Professor Heilbronner.
0002-7863/82/1504-3541$01.25/0
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ficients of the H M O secular polynomial PHMo(G)can be written by using graph theory since each term is a prescribed function of the number of bonds (edges), rings (cycles), and atoms (vertices) in the molecular g r a ~ h The . ~ polynomial ~ ~ ~ for a hypothetical cyclic resonance-free reference system Pref(G)is obtained by deleting all cyclic component terms from the original polynomial.3-6,23
pref(G)= pHM0(G)- C(ring terms)
(1)
The ordered set of the roots of PHMo(G)and P e r ( @ allow one to define GTRE as GTRE = E(PHMo)- E ( P e f )= C g i ( X i H M o - xPf) (2) I
where gi is an occupation number and the sum is over all i occupied In a polycyclic a molecular graph, it is pres~med'-'~
