J . Phys. Chem. 1986, 90, 6015-6017
6015
RISM Calculation of the Activation Barrier for Isomerlzatlon of Solvated Cyclohexane Sherwin J. Singer,+ Robert A. Kuharski,t and David Chandler*$ Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania I9104 (Received: May 19, 1986)
The solvent contribution to the activation free energy for chair-boat isomerization of cyclohexane is estimated. The rate of isomerization has been measured as a function of temperature and pressure in a variety of solvents by Hasha, Eguchi, and Jonas. Interpretation of the experimental data hinges on the behavior of the free energy. Our calculations indicate that the free energy barrier increases with increasing pressure or density of the solvent. Yet according to experiment, the isomerization rate increases with increasing pressure. These two results imply that the transmission coefficient for the isomerization of solvated cyclohexane is an increasing function of pressure, and this implication is in contradiction with recent theoretical estimates of the transmission coefficient by Zawadzki and Hynes.
I. Introduction Trapsition-state theory estimates rate constants in terms of static properties of the reactive system.' For a unimolecular activated process such as the chair-boat isomerization of cyclohexane, the effect of solvation is described by where k and ko denote rate constants in the presence and absence of solvent, respectively, 8' is Boltzmann's constant times temperature, and Ap* is the solvent contribution to the free energy of activation. That is, Ap* is the solvent contribution to the change in chemical potential or reversible work performed on the solvent in changing the C6H1,molecule from one of its chair conformers to the transition state between a chair and boat conformer:
In reality, the rate constants for isomerization differ from those estimated by transition-state theory. The differences are due to dynamical effects described by the transmission coefficient, K : k =
kTsTK
(3)
Roughly speaking, K is the fraction of successful trajectories, and its precise value can, in principle, be determined from the computation of certain well-defined time correlation functions.2 Several years ago, Montgomery et ale3suggested that for some isomerization processes in liquids, the transmission coefficient should be an increasing function of solvent density. The highpressure N M R experiments of isomerization of cyclohexane by Hasha et al.4 seem to verify this theoretical prediction. More recent theoretical work, however, calls into question the earlier prediction and the validity of the interpretation of the N M R experiments. In particular, Hynes and co-workers5 suggest that high dimensionality and vibrational energy transfer between modes of polyatomic molecules will suppress the so-called inertial or Lindemann regime of unimolecular kinetics;6 and due to that suppression, they suggest that, for cyclohexane, one will observe in a liquid only the diffusive regime first elucidated by Kramers.' The implication of this suggestion is that K is a decreasing function of solvent density. Yet, the experimental rate for cyclohexane isomerization is an increasing function of density. Therefore, if the suggestion of Hynes and co-workers is to agree with experiment, it must mean the kTST is an increasing function of solvent density, and increasing a t a rate much greater than imagined possible by Hasha et aL4 To address the issue of whether the experimental measurements of k indicate an increase or decrease in K as a function of solvent density, we have performed RISM calculations of the chemical potential surface for cyclohexane in liquid CS2for various liquid 'Address after January 1, 1987: Department of Chemistry, Ohio State University, Columbus, OH 43210. *Current address: Department of Chemistry, University of California, Berkeley, CA 94720.
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densities and for various conformational arrangements of the cyclohexane solute. Liquid carbon disulfide is one of the solvents investigated by Hasha et al." According to our calculations, the free energy of activation increases with increasing solvent density. Thus, eq 1 and 3 together with the experimentally observed rate imply that K increase$ with increasing solvent density. In the next section, we describe the required theory, and the results of our calculation are presented in section 111.
11. Calculation of Ab* Our starting point in calculating Ap* is the specification of the intermolecular interactions. We adopt the standard interaction site potential form between solvent and solute assuming a picture of the cyclohexane molecule in which the six CH2 groups are spherical extended atoms with their centers located at the carbon nuclei. A pair potential between a cyclohexane molecule and one of the solvent molecules is therefore 6
n
a=l
C s=l
um(1da) - dS) I)
Here da)denotes the position of the a t h carbon nucleus in the cyclohexane molecule, and dS)denotes the position of the sth interaction site of the solvent molecule. There are n interaction sites in a solvent molecule. With this form of interaction, the reference interaction site methodology, RISM,8 is immediately applicable. The pertinent equations for the pair correlation functions are
(4)
The symbols all have their usual meaningi8 for example, X,.(r) is the pure solvent site density pair correlation function. Equation 4 is the infinite dilution limit of the Orstein-Zernike-like equation introduced by Chandler and Andersen? and eq 5 the hypernetted-chain closure of the Chandler-Andersen equation. The (1) See, for example, Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (2) Chandler, D. J . Chem. Phys. 1978,68,2959. Grote, R. F.; Hynes, J. T. J . Chem. Phys. 1980, 73, 2700. (3) Montgomery, J. A., Jr.; Chandler, D.; Berne, B. J. J. Chem. Phys. 1979, 70, 4056. Montgomery, J. A., Jr.; Holmgren, S. L.; Chandler, D. J . Chem. Phys. 1980, 73, 3688. (4) Hasha, D. L.; Eguchi, T.; Jonas, J. J. Chem. Phys. 1981, 75, 1571. Hasha, D. L.; Eguchi, T.; Jonas, J. J. Am. Chem. SOC.1982, 104, 2290. (5) Zawadzki, A. G.; Hynes, J. T., preprint, 1985. Hynes, J. T. J . Stat. Phys. 1986,42, 149. See also, Zawadzki, A. G.; Hynes, J. T. Chem. Phys. Lett. 1985, 113, 476. (6) Robinson, P. J.; Holbrwk, K. A. Unimolecular Reactions; Wiley-Interscience: New York, 1972. (7) Kramers, H. A. Physica 1940, 7 , 284. (8) Chandler, D. Studies in Statistical Mechanics, Vol. VIII, Montroll, E. W.,Lebowitz, J. L., Ed.; North Holland: Amsterdam, 1982; p 275. (9) Chandler, D.; Andersen, H. C. J . Chem. Phys. 1972, 57, 1930.
0 1986 American Chemical Society
Singer et al.
6016 The Journal of Physical Chemistry, Vol. 90, No. 22, 1986
combination of eq 4 and 5 are the infinite dilution form of the extended RISM equation.I0 The chemical potential for the solute conformer can be obtained from the solutesolvent pair correlation functions by the standard coupling parameter integration.8 The result of this integration on using eq 4 and 5 is"
Y2 C
c,s*x,,~*cty(lr(u)- r(y)l) ( 6 )
a,Y,s,s'
where we have emphasized that psolv is a function of the solute conformation as specified by the coordinates (r(u)). Note that this dependence is found both in the explicit coordinate dependence on the right-hand side of eq 6 and also the implicit dependence due to the fact that cu8(r)and h,(r) are nonlinear functionals of ay the intramolecular pair correlation functions, (wuy(r)}. An alternative expression for pmIvis obtained from a Gaussian field theory of solvation.I2 The result from that theory is
-B~solv(lr(')l) = P
C u,s
cAr) +
72
E
~ , , * x , t * ~ ~ ~ ( -I rr(?)I) ( ~ ) (7)
U,YS,S'
Since both eq 6 and 7 are approximate, and since it is uncertain which of the two are more accurate, we present calculations using both formulas. It will be seen that the same conclusions are drawn from either expression. The solvent-induced activation free energy Ap* is obtained by subtracting psoIv((ru})calculated for the carbon positions (rut of of the transition state. Carbon-carbon the chair from pmlv({ru)) distances in the two conformers are determined from the molecular potential energy surface. W e adopt the empirical force field of Pickett and StraussI3 for the six torsional and bending degrees of freedom of the carbon ring, and treat the much higher frequency carbon-carbon stretches as frozen. The Picket and Strauss force field has an energy barrier of roughly 14 kcal/mol going from chair to boat which is about 3 or 4 kcal/mol higher than experimental estimates or more elaborate force fields which treat hydrogen atoms e~plicitly.'~ However, the conformation of transition-state cyclohexane from the Pickett and Strauss force field has not been questioned in later treatments, and we stress that our calculations only use carbon-carbon distances derived from the force field. The Pickett and Strauss reaction surface for the isomerization can be mapped onto a set of fictitious spherical angles 0 and 4.13 The reaction path for our work is determined by adjusting the torsion and bending coordinates to minimize the molecular force field energy while holding 0 and 4 fixed. This path is almost pure 0 motion, with the chair at 0 = 0, the boat at 0 = ~ / 2 and , another chair at 0 = T . The transition states occur at 0 50' and 130°, the precise value depending slightly on 4. Besides the simplified model potentials and the use of pair correlation functions from extended RISM theory to evaluate Ap*, there is another approximation implicit in our procedure. Properly, changes in high vibrational frequencies upon solvation should be accounted for. The changes are neglected, however, in our analyses of Ap* because reasonable estimates of their contribution to the free energy are no larger than a few hundredths of 8'. It may be that the total changes in Ap* are also this small, but free energetics of this size cannot effect the interpretation of the experiments reported by Hasha et al.4
-
111. Results
As stated above, the solvent we consider is carbon disulfide. The experimental equation of stateI5 is used to relate the ex~
(10) Hirata, F.; Rossky, P. J. Chem. Phys. Lett. 1981, 83, 329. Hirata, F.; Pettit, B. M.; Rossky, P. J. J . Chem. Phys. 1982, 77, 509. ( 1 1 ) Singer, S. J.; Chandler, D. Mol. Phys. 1985, 55, 621. Richardson, D. M. J. Chem. Phys. 1984,81, (12) Chandler, D.; Singh, Y.; 1975. Pratt, L. R.; Chandler, D. Methods Enzymol. 1986, 127, 48. (13) Pickett, H. M.; Strauss, H. L. J . Am. Chem. SOC.1970, 92, 7281. (14) Allinger, N. L. J. Am. Chem. Soc. 1977, 99, 8127. Schneider, H-J.; Gschwendtner, W.; Weigand, E. F. J . Am. Chem. SOC.1979, 101, 7195.
0.0
'
0
I
I
I
I
I
2
4
6
8
10
P (kbar) Figure 1. The solvent contribution to the activation free energy for cyclohexane isomerization in liquid CS2as a function of pressure. The four lines are computed as described in the text for two different models, A and B, and with two different equations for the free energy, ( 6 ) and (7).
perimental pressure, p , to the bulk density, p, employed in our calculations. We determine the pure solvent site density correlation function, xsd(r), from the extended RISM equation, employing the site-site potentials uSd(r)given by Tildesley and Madden.16 In this model each CS2molecule contains three interaction sites (of two different types) located on the respective nuclei. Tildesley and Madden's simulation studies demonstrated that their potentials satisfactorily reproduce thermodynamic and structural data of the liquid. We have compared the extended RISM calculations of the pair structure with those of Tildesley and Madden's simulation with the same potential and verified that the extended RISM correlation functions are in reasonable accord with those of simulation. For the u,(r) potentials coupling the cyclohexane solute to the solvent, we use the Lennard-Jones potential with the usual combining rules
(9) Here, the t's and u's are the energy and length parameters in the Lennard-Jones potential. The tu and ua for the extended CHI groups on cyclohexane are taken as elkB = 57.47 K and u = 3.98 A, respectively. These are the values adopted from studies of alkane liquids." The e's and u's for carbon disulfide were taken from Tildesley and Madden.I6 In addition to performing calculations with these potentials, we have also examined the case in which the full Lennard-Jones site-site potentials are replaced by the corresponding WCA repulsive sitesite potential.18 We call the interaction model with the full Lennard-Jones potentials model A, and we use model B to denote the case with the WCA repulsive potentials. Our results for AM* are plotted in Figure 1. As mentioned in the Introduction, we find that Ah* is an increasing function of pressure for both interaction models whether the free energy is estimated from eq 6 or from eq 7. Among the earlier studies of activation free energies and conformational equilibria in solution, Rossky and c o - w ~ r k e r s ' ~ have also employed the extended RISM theory and eq 6. Their results compared with simulation in a variety of circumstances have always indicated that this theory correctly predicts all qualitative trends. Further, their examination of isomerization, in particular, shows that the errors that do exist in the RISM (15) International Critical Tables, Vol. 111; McGraw-Hill: New York, 1 9 2 8 ; ' ~41. (16) Tildesley, D. J.; Madden, P. A. Mol. Phys. 1981, 42, 1137. (17) Jorgensen, W. L. J. Am. Chem. SOC.1981, 103, 335. (18) Weeks, J. D.; Chandler, D.; Andersen, H. C. J . Chem. Phys. 1971, 54, 5237. (19) Chiles, R. A.; Rossky, P. J. J . Am. Chem. SOC.1984, 106, 6867. Zichi, D. A,; Rossky, P. J. J. Chem. Phys. 1986,84, 1712. In the first of these references, the coupling parameter integration leading to eq 6 was done numerically.
J. Phys. Chem. 1986, 90, 6017-6024 theory are always in the direction of overestimating magnitudes of solvation effects. In each case examined herein, the theory predicts a small solvation effect and one that increases slightly with pressure. Therefore, in light of these observations, it seems safe to conclude that the activation barrier for cyclohexane isomerization in liquid CS2 is an increasing function of solvent pressure or density, and the corresponding transition-state theory rate constant is a decreasing function of pressure. And thus, K for this unimolecular reaction must increase with increasing pressure. W e remark that we have also examined several other simple solvents including argon modeled as a Lennard-Jones fluid. In every case
(aAk*/aP)!3 > 0
(10)
In light of our results, the study of cyclohexane in gaseous SF6 by Ross and True20 should be mentioned. This work is cited in ref 5 as experimental evidence that the Lindemann regime of the cyclohexane isomerization does not extend beyond 2-5 atm pressures, and that the increasing rate at several kilobars measured in the liquid phase by Hasha et aL4 is due to static effects. This apparent contradiction to our findings should be questioned for two reasons: First, it is not obvious that Ross and True's data (20) Ross,
B. D.; True, N. S . J . Am. Chem. SOC.1983, 105, 4871.
6017
depict the functional form assumed in ref 5-a linearly increasing rate at low pressures crossing over to a region of zero slope; other monotonically increasing functions of pressure could fit the data, and, in fact, the smallest values of d In k/dp that can be confidently extracted from the gas-phase work are all positive and at least lo2 times larger than those observed by Hasha et al! for the liquid. Second, Borkovec et al.*' have recently suggested that the Lindemann regime is not characterized by a simple linear increase of rate with pressure but displays more complicated behavior reflecting competition between time scales for energy transfer from solute to solvent and intramolecular vibrational energy relaxation within the solute. In all experiments to date,4qZ0the cyclohexane isomerization rate is observed to increase with pressure. As a result of our findings for the equilibrium contribution to the rate, the experimental observation seems best explained as a dynamical effect requiring time correlation function calculations of the transmission coefficient.
Acknowledgment. This research was supported by grants from the N I H and NSF. Registry No. Cyclohexane, 110-82-7. (21) 146.
Borkovec, M.; Straub, J. E.; Berne, B. J. J . Chem. Phys. 1986, 85,
Surface-Enhanced Resonance Raman Spectroscopy of Cytochrome c at Room and Low Temperatures Peter Hildebrandtt and Manfred Stockburger* Max-Planck-Institut fur Biophysikalische Chemie, Abteilung Spektroskopie, 0-3400 Gottingen, Federal Republic of Germany (Received: October 18, 1985; In Final Form: July 28, 1986)
Surface-enhanced resonance Raman (SERR) spectra were obtained from cytochrome c (cyt) which was adsorbed on colloidal silver particles. From a comparison of spin marker bands in the SERR spectra of cyt and the model compound iron protoporphyrin IX chloride (Fe(Cl)PPIX), excited in the Soret band, it was concluded that in cyt an adsorption-induced partial transition from the low-spin (LS) state to the high-spin (HS) state takes place. Evidence for this conversion was also found when SERR spectra were excited in the cu,P-band region at 514 nm. Spin state changes were observed for both oxidation states, cyt3+ and cyt2+. The SERR spectra provide good evidence that the specific selection rules of A-term and B-term resonance Raman scattering are maintained at the surface. By using colloidal silver in glycerol/water mixtures, we were able to determine from SERR spectra the spin-state distribution between 295 and 77 K. It turned out that cyt molecules in the two spin-state configurations exist in a thermal equilibrium which at 77 K is shifted toward the LS form. The temperature dependence of the spin-state distribution is completely reversible which indicates that cyt is not denatured at the metal surface. Spin-state changes were also reflected in the SERR spectra of various other heme proteins we had studied. The spin conversion on silver surfaces obviously is a quite general effect in heme proteins. It was ascribed to electrostatic interaction of the chromophore with the charged metal/electrolyte interface. From SERR intensities the isotherms (Frumkin-type) of cyt3+and Fe3+(C1)PPIX for adsorption on the silver sol could be determined. On this basis the enhancement factors for the two species were derived. The value for cyt3+ is 1.1 X lo6 and is a factor of 7 higher than was found for the free iron porphyrin. This implies that the position of the heme group, which is established by the protein matrix and its interaction with the metal surface, is more favorable for enhancement than the position of the free porphyrin. In terms of the electromagnetic enhancement mechanism this would suggest that the porphyrin ring in the protein lies perpendicular to the surface, while the free iron porphyrin preferentially takes in a flat position.
The discovery of the intense Raman spectrum of pyridine adsorbed on a roughened silver electrode' has opened a new branch in vibrational spectroscopy (surface enhanced Raman spectroscopy, SERS; for reviews see ref 2-4). The enhancement of the Raman scattering by several orders of magnitude (103-106) makes it possible to selectively probe the vibrational pattern of adsorbates on silver, gold, or copper surfaces (electrodes, sols, and island films). If the excitation wavelength coincides with an electronic *To whom correspondence should be addressed. Present address: Department of Chemistry, Princeton University, Princeton, NJ 08544.
transition of the adsorbate, molecular resonance Raman (RR) and SERS can combine to surface-enhanced resonance Raman ~~
~~
~
~
(1) Fleischmann, M.; Hendra, P. J.; McQuillan, A. J. Chem. Phys. Lett. 1974, 26, 163. (2) Chang, R. K.; Furtak, T. E., Eds. Surface Enhanced Raman Scattering, Plenum: New York, 1982. (3) Birke, R.; Lombardi, J. R.; Sanchez, L. A. In Electrochemical and Spectrochemical Studies of Biological Redox Components, Advances in Chemistry Series 4; Kadish, K. M., Ed.; American Chemical Society: Washington, DC, 1982; Chapter 4. (4) Chang, R. K.; Laube, B. L. CRC Crit. Rev. Solid State Mat. Sci. 1984, 12, 1.
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