928
George Nemethy and Harold A. Scheraga
in the usual Arrhenius plot of log Kobsd against T1.
Appendix I. Vibrational Energy Transfer The collection of reactive and nonreactive trajectories obtained in the Monte-Carlo rate computation contains considerable information about the energy requirements and disposition in chemical energy reactions. The usual histogram of reactions per unit of time shows a fair degree of oscillation even at the highest energy explored. The oscillation results from the fine time intervals chosen for study, and illustrates the detailed rates of mode-to-mode energy transfer. The ordinary RRKM formulation is said to presume mode to mode transfer to be very rapid relative to the dissociation rate, though Nordholm and Rice15have made clear that this presumption is not essential to the success of the RRKM theory. Here we see that the two rates are comparable. The first maximum refers to high energy in the CN bond, leading to “immediate” dissociation, the elapsed time corresponding to travel from the average value of rCNto 8ao (later maxima define characteristic times of mode to mode energy transfer). The second maximum is associated with energy transfer from the NN stretch to the reactive mode, with a lesser contribution from the bends. For all trajectories reactive in the brief period we examine, we find of course that energy in the CN stretch is productive of reaction, and that energy is drawn both from bending and NN stretching as the reaction proceeds. However, no preference is expressed concerning the initial disposition of energy in bending. Reactive trajectories producing triplet methylene are favored by high initial CN excitation, and deplete the NN stretch only slightly. In contrast, the trajectories producing singlet methylene begin with a lesser excitation in the CN stretch, and drain the NN stretch so as to overcome the higher barrier to dissociation. Bending contributes equally to singlet and triplet dissociation. Appendix 11. Revisions to QCPE 234 I t is a straightforward matter to adapt Bunker’s program13QCPE 234 to compute Arrhenius parameters
and rates for spin, or otherwise, forbidden processes. The necessary changes include the following: (1) input of parameters descriptive of the surface-to-surface coupling required in the Landau-Zener expression; (2) a sequence of statements so that rotational and vibrational partition functions can be evaluated; (3) introduction of formulae for statements quoting Arrhenius parameters; (4) branching statements to enable proper multiplication of the individual rates K ( E )by the energy-dependent Landau-Zener factor; and ( 5 ) trapezoidal-rule numerical integration of K(t) over an energy range. The changes in detail are available from C.T. (University of Virginia address) in CDC UPDATE cards or list. Acknowledgment. This work was supported in part by NATO Grant 1186 and the University of Virginia Sesquicentennial Foundation.
References and Notes (1) (2) (3) (4) (5)
(6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
D. W. Setser and B. S. Rabinovitch, Can. J. Chem., 40, 1425 (1962). M. Y. Chu and J. S.Dahler, Mol. Phys., 27, 1045 (1974). J. C. Tully, J . Chem. Phys., 61, 61 (1974). K. C. Kulander and J. S. Dahler, Chem. Phys. Left., 41, 125 (1976). E. A. Halevi, R. Pauncz, I.Schek,and H. Weinstein in “The Jerusalem Symposia on Quantum Chemistry and Biochemistry”, Vol. 6, E. D. Bergmann and B. Pullman, Ed., The Israel Academy of Sciences and Humanities, 1974, pp 167-182, but see the following discussion and L. Salem and C. Rowland, Angew. Chem., Int. Ed. Engl., 11, 92 (1972); F. S. Rowland, D. S. T. Lee, D. C. Montague, and L. Russell, Discuss. Faraday Soc.,53, 111 (1972); R. F. W. Bader and J. I. Generosa, Can. J . Chem., 43, 1631 (1965); B. S. Rabinovitch and D. W. Setser, J. Am. Chem. Soc., 83, 750 (1961). J. C. Tully and R. K. Preston, J. Chem. Phys., 55, 562 (1971). D. L. Bunker, J. Chem. Phys., 37, 393 (1962); 40, 1946 (1964). R. Beckett and J. Hurt in “Numerical Calculations and Algorithms”, McGraw-Hill, New York, N.Y., 1967, Chapter 6. C. B. Moore and G. C. Pimentel, J. Chem. Phys., 40, 329 (1964); 40, 342 (1964). C. Zener, Proc. R. Soc. London, Ser. A , 137, 696 (1932). W. Forst, “Theory of Unimolecular Reactions”, Academic Press, New York, N.Y., 1973. For example: P. Lechkten, R. Breslow, A. H. Schmidt, and N. Turro, J. Am. Chem. Soc., 95, 4765 (1973). D. L. Bunker, Program 234, Quantum chemistry Program Exchange, Bloomington, Ind. R. T. Conley, “Infrared Spectroscopy”, 32nd ed, Allyn and Bacon, Inc., Boston, Mass., 1972. K. S. J. Nordholm and S. A. Rice, J . Chem. Phys., 61, 203, 768 (1974); 62, 157 (1975).
Intermolecular Potentials from Crystal Data. 5. Determination of Empirical Potentials for O-H***O Hydrogen Bonds from Packing Configurations and Lattice Energies of Polyhydric Alcohols’ George Nimethy and Harold A. Scheraga”* Department of Chemistry, Cornel1 University, Ithaca, New York 14853 (Received January 3, 1977)
Crystal packing computations of four polyhydric alcohol molecules were used to determine the values of the parameters for the potential energy of the 0-H4--OI8hydrogen bond, to be used in a self-consistentset of potential energy parameters, employed in peptide conformational energy computations. The computed lattice constants and energies are in good agreement with the experimental values.
I. Introduction Computations on the conformations of polypeptides and proteins are carried out in this laboratory by using a set of empirical energy parameters developed to describe The Journal of Physical Chemistry, Vol. 81, No. 9 , 1977
interatomic interaction^.^,^ The potential functions for nonbonded interactions and for hydrogen bonds were parameterized by minimizing the potential energies of suitably chosen crystals and fitting the computed lattice constants to the observed value^.^ In this paper, we report
929
Intermolecular Potentials from Crystal Data
i‘
A.
“‘I
t
-320/
H4 470
C
D.
f
,188 14 ~ 3 4 2 13
2
\ -325
,
\
152 H4
Y
“171
t Figure 1. The structures and CNDO/P (ON) partial atomic charges indicated above the atomic symbols in electronic units (X 1000) for polyhydric alcohol molecules: (A) meso-erythritol, (6) a-bglucose, (C) pentaerythritol, (D) glycerol. The conformations shown are those occurring in the respective crystals. The positions of the hydrogens in the hydroxyl groups are those which correspond to best intermolecular hydrogen bonding in the energy-minimized structure. The subscripts designate atom types as defined in Table I of ref 5.
a revision of the hydrogen-bonding parameters for one of the types of hydrogen bond considered earlier.5 We obtained the parameters for the general hydrogen bond potential between a hydroxyl or carboxylic acid hydrogen and a hydroxyl or carboxylic acid (C-0-H) oxygen. These are designated as atom types 4 and 18, respectively (cf. Table I of ref 5). The bond is referred to as an H4--OI8hydrogen bond.5 The parameters for this hydrogen bond were determined indirectly in the earlier in contrast to the parameters for other hydrogen bonds. In the present study, we determined these parameters by fitting the crystal structures of four polyhydric alcohols. Structures were chosen in which the intermolecular interactions are dominated by H4-OI8 hydrogen bonds.
11. Methods Interaction between the H4 and OI8 atoms is represented5f by a general hydrogen bond potential (GHB) of the form UGH,= A)H ...x/rH ...xl*- BH ...x/rH ...x10
(1)
where X is the acceptor atom, in this case OI8. The position and depth of the minimum of this potential are designated as rmhand Umh,respectively. It must be noted that U G H B represents only part of the total energy of the hydrogen bond. The total energy includes electrostatic interactions as well as nonbonded interactions with other neighboring atoms.5 The parameters for nonbonded interactions were taken from the previous s t ~ d y . The ~ The Journal of Physical Chemistry, Vol. 81, No. 9, 1977
George Nemethy and Harold A. Scheraga
930
partial charges on the atoms were determined by a molecular orbital CNDO/2 (ON) cal~ulation.~They are shown in Figure 1. The value for the attractive parameter, B H ...X = 4610 kcal 81°/mol, was taken from an earlier study.6 The repulsive coefficient, was adjusted by minimizing the binding energy of the crystals, as done previous1y.j The method used was that reported in the previous s t ~ d y . The ~ coordinates of the heavy atoms of each molecule were generated from the known molecular structures.8-12 The positions of the hydrogen atoms in a-D-glucose are known from a neutron diffraction study.l0 For the other molecules, the hydrogens in CH2 and CH groups were generated by adopting a standard geometry, as described earlier (Section IID of ref 5 ) . The hydroxyl (-OH)groups were first rotated into a position which corresponds to the hydrogen conformation of lowest intermolecular energy. The molecules in the unit cell and in neighboring unit cells were generated by considering the molecules as rigid b ~ d i e s The . ~ crystal lattice energy was then minimized with respect to the unit cell lattice constants a, b, and c. In all four crystals, a = /3 = y = 90°; these lattice parameters were kept constant. The optimal value of was obtained as the one which minimized the root-mean-square deviation u of the lattice parameters: (z
=
c
(1/3)
3
x(l;alcd i= 1
- i.obsd I
(2)
2
where 1, = a, b, c, respectively. This procedure was carried out for every crystal separately. The final value of is the one which minimizes the sum of the u's for all four crystals. 111. Results and Discussion
Optimal fitting was obtained with a choice of Ak...x = 11220 kcal 812/mol. This value, taken together with the previously established5p6value of B = 4610 kcal Alo/mol, results in r,,, = 1.71 8 and U,,, = -3.61 kcal/mol for the GHB potential of the Hp-O18hydrogen bond. Computed crystal parameters are compared with the experimental values in Table I. For three of the crystals, the deviation u, taken as a function of A h...x,has a minimum at the value of A" ...x cited. The minimum is a deep one for meso-erythritol and for glycerol, but is less sharply defined for a-&glucose. For pentaerythritol, the value of u according to eq 2 is relatively high (Table I). It is a slowly varying function of Ak,x and has no minimum. However, it must be noted that the pentaerythritol molecule is linked to its neighbors in the crystal by hydrogen bonds in the X-Y plane only, forming a two-dimensional network.'l The layers, stacked along the x axis, interact by means of weak nonbonded interactions. The high negative value of Ac found for this crystal suggests5that thermal expansion increased the unit cell size in the z direction in the crystal structure, which was determined at high temperatures, more than in the x and the y directions. The observed amplitude of thermal vibrations of both C and 0 atoms was reported to be highest in the z direction as well.ll Thus, thermal expansion in the observed crystal can account for the increased deviation of Ac seen here. Ac is not sensitive to changes in the GHB for the H,.-O18 bond. In fact, if one recalculates B, based on Aa and Ab alone, it has a minimum at A'H,..~= 11170 kcal 812/mol, with u = 0.03 A. As seen from Table I, the calculated and observed lengths of the hydrogen bonds agree well. The bond lengths are very near the optimal value of rmin= 1.71 8, except in glucose where the requirement of packing the The Journal of Physical Chemistry, Vol. 87,
No. 9 , 1977
I
CD
-
d
0 I
..
.^ m
0
d 31
I
Reaction of ea;
+ H30+
931 (PCM75-08691). (2) To whom requests for reprints should be addressed. (3) F. A. Momany, R. F. McGuire, A. W. Burgess, and H. A. Scheraga, J . Phys. Chem., 79, 2361 (1975). (4) These computer programs and their description, and all associated geometric and energy parameters, are available on magnetlc tape from the Quantum Chemistry Program Exchange. Write to Quantum Chemistry Program Exchange, Chemistry Department, Room 204, Indiana University, Bloomington, Ind. 47401 for standard program request sheets, and then order No. QCPE 286. (5) F. A. Momany, L. M. Carruthers, R. F. McGuire, and H. A. Scheraga, J. Phys. Chem., 78, 1595 (1974). (6) R. F. McGuire, F. A. Momany, and H. A. Scheraga, J. Phys. Chem., 76, 375 (1972). (7) J. F. Yan, F. A. Momany, R. Hoffmann, and H. A. Scheraga, J. Phys. Chem., 74, 420 (1970). (8) A. Shimada, Bull. Chem. SOC. Jpn., 32, 325 (1959). (9) T. R. R. McDonaldand C. A. Beevers, Acta Crystallogr.,5,654 (1962). (10) G. M. Brown and H. A. Levy, Science, 147, 1038 (1965). (11) R. Shiono, D. W. J. Cruickshank, and E. G. Cox, Acta Crystallogr., 11, 389 (1958). (12) H. van Koningveld, Red. Trav. Chim. Pays-Bas, 87, 243 (1968). (13) I. Nitta, S. Seki, M. Mamotani, K. Suzuki, and S. Nakagawa, Proc. Jpn. Acad. Scl., 26, 11 (1950). (14) R. S. Bradley and S. Cotson, J. Chem. Soc., 1684 (1953).
inflexible molecules may prevent optimal hydrogen bonding approaches of H4and OI8. The hydrogen bond is nearly linear, with the 0-H-0 angle above 162' for all bonds, and above 170° in most cases. The calculated lattice energies and experimental sublimation energies agree fairly well. The calculated energies are smaller in magnitude than the observed values. However, the latter may contain contributions from sources which are not taken into account in these computations, such as vibrational terms. Acknowledgment. We thank Drs. L. G. Dunfield and F. A. Momany for helpful discussions and Marcia Pottle for aid with the computer programs. References and Notes (1) This work was supported by grants from the National Institute of Aging and from the National Institute of General Medical Sciences, of the National Institutes of Health, US. Public Health Service (AGO0322 and GM-14312), and from the National Science Foundation
The Reaction of ea; 4- H30f. Concentration Effects of Acid or Salts C. D. Jonah, J. R. Miller, and Max 8. Matheson" Chemlstry Division, Argonne Natlonal Laboratory, Argonne, Illinois 60439 (Received January 17, 1977) Publication costs assisted by Argonne National Laboratory
The rate constant, k(e,; + H30+),has been measured in dilute HC104,alone and with LiC104added up to 2.5 M, and in concentrated HC104,alone and with LiC104or NaC104added up to 2.5 M. The rate constant has the same dependence upon total concentration either for HC104alone or for HC104+ LiC104. The results show that changes in k(e,; + H30*) are determined by changes in the diffusion-controlledencounter rate and by changes in the 'activities of e,, and H30+. Any time-dependent effect in the rate constant is not more than 10% for hydrated electrons with lifetimes longer than 30 ps.
Introduction The rate constant for the reaction ea&
+ H,O+-+H t H,O
(1)
has been measured in dilute solution by a number of workers,' with a preferred value for kl of 2.3 X 10" M-l s-'. Although, this is a rapid reaction it is still only about one fifth the diffusion-controlled encounter rate: the latter being very high owing to the high diffusion constants of eaq-and, especially, H30+. In the presence of added salts up to ionic strength about 0.05, kl for dilute acid obeys the Bronsted-Bjerrum equation
as shown by Czapski and S~hwarz.~ k I ois the value at ionic strength, p, equal zero and the 2 s are the units of electronic charge on e a i (-1) and on H30+ (+I). In concentrated solutions of HC1 or HC104 Hunt and his colleagues4used his stroboscopic pulse radiolysis technique to determine kl and found k1 varied from 1.0 to 1.4 X 1O1O between 0.3 and 5.0 M acid with a shallow minimum at about 1 M. Schwarz5 interpreted these results in terms of ionic strength effects (activity coefficients) plus time
dependent effects at the highest concentrations (shortest times). We have remeasured kl in both dilute and concentrated perchloric acid, with and without added salt. In this paper we present these results and from them deduce that activities and encounter rates control reaction 1 and that there are no important time-dependent effects at times greater than 30 ps. Experimental Section Apparatus. For experiments with dilute acid the Argonne ARC0 electron linac was used with standard pulse radiolysis techniques involving photomultiplier detection! Electron pulse length was set at 4 ns. A pulsed xenon lamp was used as the analyzing light for measurement of e, - at 600 nm. Cell length was 5 cm with one light pass. bell filling techniques have been previously r e p ~ r t e d . ~ Rate constant measurements at high acid concentrations were made with the Argonne stroboscopic pulse radiolysis apparatus,8 which has a time resolution of about 40 ps. The microwave linac is pulsed 60 times per second, each pulse consisting of a main pulse containing more than 90% of the charge and with a width (fwhm) of 30 ps. The beam is spread horizontally, and about 30% in the center is intercepted by a Cerenkov generator (contains 1atm of xenon), while the rest of the beam is passed around a 270° The Journal of Physical Chemistry, Vo/.81, No. 9 , 1977
