NOTES
1431
crystal were apparently quenched in the pyrene-doped fluorene crystal. All of the above evidence supports the conclusion that the e.s.r. spectra observed in the pyrene-doped fluorene crystals arise from the (lowest) triplet state of pyrene. Further splitting of the fine-structure e.s.r lines into hyperfine lines was not observed in any of the spectra recorded,11 and, therefore, it is not possible to determine uniquely the orientations of the pyrene molecules in the fluorene lattice. However, one of the principal magnetic axes of pyrene lies approximately along the crystalline c axis (the long axis of fluorene), and the other two magnetic axes correspond to the short axis and a normal to the molecular plane of fluorene. It appears, therefore, that the molecular plane and long axis of the pyrene molecule are oriented with respect to the fluorene lattice in the same manner as the displaced fluorene molecule. If we assume this to be the case, then the molecular z, y, and x axes of pyrene would be parallel to the long axis, the short axis, and the normal to the molecular plane of the pyrene molecule, respectively. l 2
Ackno wbdgments. We are indebted to Professor Harden M. McConnell for the generous use of his laboratory facilities. We are also indebted to Dr. Richard E. Marsh for help concerning the X-ray photography, to Professor Melvin W. Hanna for helpful discussions, and to Xrs. Lelia M.Coyne for obtaining the optical spectra. (11) The lack of resolution is presumably due to the large number of protons interacting with the electron and to crystal disorder. Laue photographs disclosed that the first crystals obtained were badly disordered. The crystals used for the e.s.r. study were of much better quality but could have been sufficiently disordered to reduce resolution of the hyperfine structure. D and E for pyrene were arbitrarily chosen (12) By to have opposite signs. If D and E were chosen to have the same sign then the 2 and g axes would be interchanged. This indeterminancy affects only the relative signs of D and E and not their magnitudes. We are indebted to Dr. J. H. van der Waals for a discussion on this point.
Complexions' A n Exact High Speed Method Of
"'*
by E. W. Schlag, R. A. Sandsmark, and W. G. Valance ~
Department of Chemistry, A~orthwestern UnheTsity, Evanston. Illinois (Receitied Nonember 6 , 196.4)
There has been a substantial increase in interest in obtainirlg information about kinetic rate processes on a
molecular level. For such experiments it is necessary that one is able to define initial states as closely as possible in order to be able to discriminate between various possible theoretical models on the basis of the experimental results. One such approach has been to study the reactions of monoenergetically excited species.' One then wishes to compute a set of theoretical transition probabilities for reaction or energy transfer and then use this set to generate the appropriately predicted laboratory observable.2 If the set of transition probabilities involves any statistical factors, then one needs to know the number of states that all have a total energy between E and E dE, known simply as the density of states p(E) a t E . For example, in the RRKM version of unimolecular rate theory, the transition probability for reaction at E is just proportional to a simple ratio3 of two such numbers, that for an activated complex (at a reduced E ) to that for excited molecules at E. Similar considerations may arise for some theories of energy transfer.2 Hence, ideally, one would like to generate a set p(E) for all E of interest, for all relevant species. The primary information for such a computation would just be the frequency spectrum of each of the species. In detail, the generation of such a complete set of p(E) just amounts to a computation which takes the molecule to be a collection of nonconimensurate and perhaps anharmonic oscillators. The approximation that considers all oscillators to be identical leads to serious inaccuracies.* One can, however, simplify - the probleni by neglecting anharnionicity effects, which may often be a justifyable approximation,j There has been Some question as to the formula to be employed in the computation of this density at E. ~i~~ and Marcus3 presented a modification Of the for use a t higher energies. (At lower energies they suggested a count Of possible complexions') Rabinovitch and Diesens Dointed to inadeauacies in this expression for typical high energies of kinetic
+
(1) For examples, see: (a) B. S. Rabinovitch and 31. C. Flowers. Quart. Rev. (London), 18, 122 (1964); (b) G. B. Porter and B. T. Connelly, J . Chem. Phys., 33, 81 (1960); (c) A. N.Btrachan, R. K. Boyd, and K. 0. Kutschke, Can. J . Chem., 42, 1345 (1964). (2) For a discussion of the various methods of calculating laboratory observables from transition probabilities and their difficulties, see R. V. Serauskas and E. W. Schlag, J . Chem. Phys., in press, and references cited therein. (3) R. A. Marcus and 0. K. Rice, J . Phus. Colloid Chem., 5 5 , 894 (1951). (4) This can be seen from the expression developed by L. S. Kassel for nonidentical harmonic oscillators, ibid., 32, 1065 (1928). ( 5 ) See E. W. Schlag, R. A. Sandsmark, and W. G. Valance, J. Chem. Phys., 40, 1461 (1964), for a typical evaluation of the anharmonic contributions to the density of states. (6) B. S. Rabinovitch and R. W. Diesen, ihid., 30, 735 (1959). See also G. 2. Whitten and B. 9. Rabinovitch, ibld., 3 8 , 2466 (1963).
Volume 69, .\Tumber 4
April 1965
NOTES
1432
interest, arid introduced an empirical correction factor. Schlag and Sandsmark' suggested a nonempirical finite series, which had a leading term identical with the single term expression given by Marcus and Rice.3 Further terms become important a t lower energies. Schlag and Sandsmark, however, had developed only the first three terms of their series. More recently, further terms in this series were developed by an inverse Laplace transform expansion technique.*, In particular, Thiele has given a general coefficient'O from which, as he has shown, the first four terms of the series of Schlag and Sandsmark can be generated by a multinomial formula. In view of the apparent usefulness of an exact and simple method for generating p(E), it was decided to generate the complete series for several molecules from this multinomial formula and compare this method of counting with a detailed combinatorial calculation.' This was programmed on a computer since the collation of terms becomes an impossible bookkeeping task. The general term in the series is generated by a proper collection of Thiele's specific terms. The number of terms in this series, when complete, is approximately half the nun~berof vibrational degrees of freedom. The complete series was computed for methyl chloride, cyclopropane, cyclopropane-&, CBr3C1,CClzO, CHBr3, C2H4,C4H2, and benzene." The typical computation time was about 1 min. to calculate a complete set of densities, p(E), for all E up to 250 kcal. When one compares the densities obtained from the complete series with exact combinatorial cal~ulations,~ one finds only small discrepancies for these molecules at 10 kcal. and above, at most lo%, the discrepancy being further reduced at higher energies (see Table I). Below this energy, the densities fluctuate more strongly about the average value given by the series. This causes no difficulty since it takes but a few seconds to compute the values up to 10 kcal. by the exact combinatorial n~ethod.' Hence, one can reduce the exact
Table I ,Molecule
Methyl chloride Cyclopropsne C yclopropane-ds CBrsCl CClZO CHBra
C2Ha C4Hl Benzene
-Discrepancy 10 kcal.
at (in %)-15 kcal. 20 kcal.
8.16 -9.45 3.70 -0.01 0.88 1.37 0.32 -3.80 3.86
-0.52
The Journal of Physical Chemtetry
-3.47 -0.33
-0.02 -0.19 -0.01 0.00 1.50 -1.93
-1.73 -2.00 -0.80 -0.01 0.04 -0.22
-1.42 -1.70
calculation of densities to a very high speed computer subroutine. For an arbitrary group of frequencies, the subroutine will generate the value for all densities, p(E), up to 250 kcal.12
Acknowledgment. The support of this work by a grant from the Petroleum Research Fund of the American Chemical Society is gratefully acknowledged. (7) E. W. Schlag and R. A. Sandsmark, J . Chem. Phys., 37, 168 (1962). (8) (a) P. C. Haarhoff, Mol. Phys., 6,337 (1963); (b) P. C. Haarhoff, ibid., 7, 101 (1963). (9) E. Thiele, J . Chem. Phys., 39, 3258 (1963). (10) It should be pointed out that these are only the coefficients for the expansion about the pole p = 0. The effect of higher poles can be shown to just lead to oscillations on top of this smoothed function, without changing its average value as a function of energy. The effect of these higher poles will only be important for the first few kcal./mole. Computationally, this is adequately handled by a simple direct count up to 10 kcal./mole (see below). The Dirichlet integral technique of Schlag and Sandsmark effectively also rejects these oscillatory terms when it replaces a multiple sum by a multiple integral. Since only the first 10 kcal./mole are involved, a direct count procedure is to be preferred to explicitly including quantum corrections in the series techniques. (11) Frequencies chosen were those listed in ref. 6. (12) A copy of the program written in IBM Fortran I V (IBSYS system) with instructions is available upon request.
Comparison of Water Sorption and Deuterium-Hydrogen Exchange Sites in Poly -L-valine l a
by W. W. Brandt and R. S. BudrysIb Department of Chemistry, Illinois Institutc of TechnoEogy, Chicago, Illinois 60616 (Received November SO, 1964)
Measurements of deuterium-hydrogen (D-H) exchange capacities and rates on dissolved polypeptides or proteins have become one of the standard methods to determine the chain conformations of these macromolecules.2 H atoms which are readily exchanged are usually assumed to be easily accessible. In solid polymers, on the other hand, D-H exchange capacities may (1) (a) This work was supported by PHS Grant AM-4324. Material supplementary to this article has been deposited as Document No. 8130 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C. A copy may be secured by citing the document number and by remitting $2.50 for photoprints, or $1.75 for 35 mm. microfilm. Advance payment is required. Make checks or money orders payable to: Chief, Photoduplication Service, Library of Congress. (b) Abstrated in part from the work of R. S. B. to be submitted in partial fulfillment of the research requirements for the Ph.D. degree in the Department of Chemistry, Illinois Institute of Technology. ( 2 ) H. A. Scheraga, "Protein Structure," Academic Press, Inc., New York, N. Y.,1961.
