J. Phys. Chem. 1981, 85, 3974-3978
3974
sion, and perhaps might also change the barrier to surface or interfacial steps. Whether or not the sensitivity of flux to sample history is neglected, it is apparent that in the pressure range from Pco2< PL cr: to Pco N 10-2Peq,where the C02flux against the sample exceed2s the decomposition flux by a factor of -lo3, the flux is essentially independent of Pco,. Thus, in this regime, the molar flux density J is given by an equation of the form J = k(Peq- Pco,) N kPeq (1) The model of Searcy and Beruto yields this expression for the pressure dependence when the rate-limiting process is either solid-state diffusion of C02 or a surface step for CO2p That model did not consider the possibility that the solid component of a decomposition reaction might be displaced from the reactant phase to the product solid phase by a cooperative process-such as twinning or shear-in which groups of atoms or ions move simultaneously. In CdC03 decomposition, the CdO phase probably forms by twinning in volume elements of the CdC03 after diffusion of all or part of the C02from those elements makes them sufficiently unstable.1°J3J4 A similar process can be expected in CaC03 decomposition.1°J3J4 If so, because cooperative processes are very rapid, the rate-limiting process must be diffusion of C02 from the CaC03volume elements near the advancing reaction front or a surface step in the desorption of C02, and the predicted rate equations then have the same form as eq 1: At low C02 pressures, condensed-phase diffusion of COz (perhaps as countercurrents of CO2- and 02-ions), or a surface step of C02,e.g., decomposition of C032-to 02and adsorbed C02, is the rate-limiting process in CaC03 decomposition.1° The possibility that the final step of C02 desorption is rate limiting can be eliminated by comparison of the ap(13) J. C. Niepce and G. Watelle, J. Muter. Sci., 13, 149 (1978).
(14) C. L. Cronan, F. J. Micole, M. Topic, H. Leidheiser, A. C. Zettlemoyer, and S. Popovic, J. Colloid. Interface Sci., 55, 546 (1976).
parent activation entropy, calculated when desorption is assumed to be rate limiting, to the standard entropy of the reaction CaC03(s) = CaO(s) C02(g). Even if CaO is produced in a thermodynamically active state, its entropy would be unlikely to increase by more than 8-12 J/(mol deg) over the entropy of normal CaO. By analogy with an analysis given elsewhere for congruent vaporization,15if desorption of C02 were rate limiting the transition state would have properties close to those of the separate CaO solid phase plus C02 gas, and the apparent entropy of activation should be close to the entropy of the equilibrium decomposition reaction. The apparent entropy of activation AS* can be calculated from the measured apparent activation enthalpy4 and the values of PLin Figure 2 by means of the relation In PL= -AH*/RT AS*/R. If PL from ref 4 at 1000 K is used, AS* is calculated to be 84 f 12 J/(mol deg) (20 f 3 cal/(mol deg)),which is much lower than the standard entropy, 147 J/(mol deg) (35 cal/(mol deg)) at 1000 K. The present study shows that the mechanism for calcite decomposition is different at Pco,> 10-2Peqthan in the low-pressure range, but the dependence on Pco,at high C02pressures is not parabolic as concluded from earlier studies over more limited pressure ranges. Shukla and Searcy will report a more detailed study16of the influence of high relative C02 pressures on calcite decomposition rates.
+
+
-
Acknowledgment. D. J. Meschi, James A. Roberts, Jr., D. Beruto, Rama Shukla, T. K. Basu, and Gary Knudsen provided advice on various aspects of this study. This MS thesis study of Taghi Darroudi was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy under Contract No. W-7405-ENG-48. (15) A. W. Searcy in "Chemical and Mechanical Behavior of Inorganic Materials", A. W. Searcy, D. V. Ragone, and U. Colombo, Ed., WileyInterscience, New York, 1970, Chapter 6. (16) R. Shukla, Ph.D. Thesis, Lawrence Berkeley Laboratory, Dec 1979.
Franck-Condon Model for Collinear Reactive Systems. Factorization of the Reactive Vibrational Amplitudes and Probabilities Michael Baer' Department of Physics, Universlty of Kalserslautern, 6750 Kalserslautern, Federal Republlc of &rmany I n Final Form: August 11, 198 1)
(Recelved: May 15, 198 1:
This work presents a model for collinear reactive scattering which, under certain conditions, becomes a Franck-Condon-type model. Further analysis leads to the factorization of the reactive vibrational amplitudes (and probabilities) which yields two linearly dependent variables for each final vibrational state. From the parameters of the straight line, the ratio of the two fundamental frequencies of the oscillators and their relative coordinate shift can be obtained, The model is applied to the two isotopic reactions H(D) + Clz H(D)Cl
-
+ c1.
Introduction The Franck-Condon (FC) model for collinear reactive scattering and the two isotopic reactive systems H(D) + C12 H(D)Cl + C1 (1)
brational distributions obtained for these systems led to the FC mode1.l Since then, the FC model has been applied in different forms to this and other systems, with varying degrees of and the H + C12system has
-+
are closely related; indeed, the quantum-mechanical vi0022-3854/81/2085-3974$0 1.25/0
(1) M. Baer, J. Chem. Phys., 60, 1057 (1974).
0 1981 American Chemical Society
Franck-Condon Model for Collinear Reactive Systems
The Journal of Physical Chemistry, Vol. 85, No. 26, 1981 3975 +0.4
r
0
Reactants
-0 4
L I
Reaction coordinate Flgure 1. Hirschfelder-Wlgner model. The tubes for reagents and products are matched at some value along the reaction coordinate, to form one single Itnear tube.
served as a test case for many models and approaches.2-18 In the present work, the FC model is reconsidered, but from a different viewpoint. We start by modifying a model, originally devised by Hirschfelder and Wigner,lg to account for strong vibrational nonadiabatic transitions and which, under certain conditions, leads to FC-type transition probabilities. Next, we apply the analytic expression of KatrielZ0for the overlap integral between two shifted harmonic oscillator wave functions, to show the existence of the factorization of the vibrational amplitudes and the corresponding probabilities. (This result is reminiscent of a similar feature obtained within the framework of the infinite order sudden approximation for rotational transitions in inelasticz1and reactive scattering.22) Finally, based on the linear relations of the reactive amplitudes from the ground state and from the first excited state, two linearly dependent variables for each final vibrational state are found. The parameters of the line yield the ratio between the two fundamental frequencies and the amount of the relative shift. The approach is applied to the reactions given in eq 1. One of the main aims of this study is to construct analytical tools by which we could determine unambiguously the extent to which FC-type processes really exist in chemical reactions.
Hirschfelder-Wigner Model In the Hirschfelder-Wigner modePg the two skewed
L
>
*
*
8
6
0 5 025 0 025 05
r (a.u.1
Flgure 2. Relative position of the two harmonic potentials at the matching points. Maximal positions of the wave functions at each oscillator are shown as are the possible Franck-Condon-type transitions.
reactive channels are lined up and in this way a straight linear tube, along a coordinate R, is formed (see Figure 1). The two channels are characterized by their fundamental frequencies, w and Q,respectively. In case the reaction is exothermic (endothermic), one channel is shifted upward along the energy axis by an amount AV, to account correctly for the exothermicity. In this form, the model is appropriate for reactions which are mainly vibrationaladiabatic, e.g., the H + H2 system; however, it can hardly account for vibrational-nonadiabatic processes, irrespective of the size of AV. To incorporate nonadiabaticity, one should also shift the two tubes, one with respect to the other, along the vibrational coordinate. The relative positions of the two oscillators at the matching point are shown in Figure 2. Assuming the line that separates the reagent channel from the product channel is R = 0, then the asymptotic solutions for the two channels are rL(r,RR) = N
4f,,(r) + C (kf,a/Ki)1'2R,n,eik~R4f,(r)R 1 0 n=O
(2)M. J. Berry, Chem. Phys. Lett., 27,73 (1974). (3)U.Halavee and M. Shapiro, J. Chem. Phys., 64,2826 (1976). (4)G. C. Schatz and J. Ross, J. Chem. Phys., 66, 1021,1037,2943 (1977). (5)S.Fischer and G. Venzl, J. Chem. Phys., 67, 1335 (1977). (6)J. K.C. Wong and P. Brumer, Chem. Phys. Lett., 68,517(1979). (7)M. S.Child and K. B. Whaley, Faraday Discuss. Chem. SOC.,67, 57 (1979). (8)M. Baer, J. Chem. Phys., 62,4545 (1975). (9)H. Essen, G.D. Billing, and M. Baer, Chem. Phys., 17,443(1976). (10)D. G.Truhlar, J. A. Merrick, and J. W. Duff, J.Am. Chem. Soc., 98,6771 (1976). (11)M. Baer and J. A. Beswick, Chem. Phys., 21,443 (1977). (12)U.Halavee and R. D. Levine, Chem. Phys. Lett., 46,15 (1977). (13)J. N.L. Connor, W. Jakubetz, and J. Manz, Chem. Phys., 28,219 (1978). (14)J. N.L. Connor, A. Lagana, J. C. Whitehead, W. Jakubetz, and J. Manz, Chem. Phys. Lett., 62,479 (1979). (15)M. V. Basilevsky and V. M. Ryaboy, Chem. Phys., 41,461,477, 489 (1979). (16)J. C. Gray, D. G. Truhlar, and M. Baer, J. Phys. Chem., 83,1045 (1979). (17)E. Pollak and R. D. Levine, J. Chem. Phys., 72, 2484 (1980). (18)J. N.L. Connor, W. Jakubetz, J. Manz, and J. C. Whitehead, J. Chem. Phys., 72,6209 (1980). (19)J. 0.Hirschfelder and E. P. Wigner, J. Chem. Phys., 7,616(1939). (20)J. Katriel, J. Phys. B, 3, 1315 (1970). (21)R. Goldflam, S.Green, and D. J. Kouri, J. Chem. Phys., 67,4149 (1977). (22)D. J. Kouri, V. Khare, and M. Baer, J. Chem. Phys., 75, 117-9 (1981). ~
$(r,R) =
M
(ki,/kII)l/ZT m nnoe-ikrn"R 4 m 11 ( r )
R
50
(2)
m=O
Here R,,, and T,,, are the reflected and transmitted coefficients (nobeing the initial state), 4 i ( r ) and 4 i ( r ) are the vibrational eigenfunctions for the two channels, and k', and k i are the corresponding wavenumbers: k f ,= ( ( 2 p / h 2 ) [ E - (n + f/z)hQ])1/2 n = 0, 1, ...,N
k z = ( ( 2 ~ / h ~ -) [(rn E + f/2)hw])1/2 m = 0, 1, ..., M (3) (Only open states are considered below.) Here p is the reduced mass, E is the energy, and N and M are the numbers of open states in each channel. Matching these two solutions and their derivatives at R = 0 leads to the following system of algebraic equations for Tmnoand R n 4 N
T,,, - C&LRnnO= CT$& n
m = 0, 1, ...,M (4)
M
C a ~ ~ T , , ,+, , R,,, , = ,6 m
where
n = 0, 1, ..., N
3978
Baer
The Journal of Physical Chemistry, Vol. 85, No. 26, 1981 Q%
= (kK/kk)'I2Smn (2) Qnm
Smn
=
=
(hnl+n I1 I )
(5)
(1) Qmn
(6)
Thus, Smn is an overlap integral between two shifted harmonic oscillator wave functions. Let us now consider the case where no reflection takes place, i.e., R,, = 0 for all n. Then it can be seen from the first M + 1 equations in eq 4 that
T,,
= agio
m = 0, 1, ..., M
= (k~/k~J1Smn,12 m = 0, 1, ..., M
(8)
Among other things, the second system of equations in eq 4 guarantees the unitarity (assuming R,,, 0), as can be seen by considering the case n = no and substitution for ,,Q,:,! and Tmn: M
M
M
CQL%Tmn, = Z(kE/k&)lSmn,J2 = C P m n , = 1 m
m
Sm+lO
=
Now, introducing
x and r
defined as
(7)
which yields the following for the corresponding reactive transition probabilities Pmno:
P,,
which, following the substitution for the overlap integrals, becomes
m
leads to Sm+lO
=
Here, r is the relative shift measured in units of half the width of the harmonic oscillator at the ground state and x is the ratio between the fundamental frequencies of the two oscillators. The Case Where n = 1.
In the following, eq 8 is assumed to be valid, and for a given Pmnwe therefore have S m , = *[(k~o/k~)pmno11/2
(9)
(the overlap integrals are assumed to be real).
Factorization of the Reactive Vibrational Amplitudes and Probabilities The analytic form for the overlap integral S, between two shifted harmonic oscillator wave functions was given by KatrieL2O In order to write it in a convenient form, the following notations are introduced: A =( 2 E )
112
E)
112
q = 2b(
t = - -h b2p
=
Jmo
-
Q
n+w
(10)
or, again applying eq 16, it takes the following form:
Q + W
QW
where b is the amount of the shift. Thus S,
(19) tJm-10 This equation, once the Jmn'sare replaced by Smn's,becomes Jm1
exp(-1/2t)
112
P=-2b(;)
The first summation is recognized as Jmo and the second as Jm-lo. Thus
= Am!n!pmqnJm,
(11)
where
Based on this result, two special cases are considered. The Case Where n = 0.
Following some algebraic rearrangements, one may show the existence of the recursion relation
Equation 21 is the reactive vibrational factorization for the transitions from the first excited vibrational state in terms of the corresponding amplitudes for transitions from the ground state. There seem to be no essential difficulties in obtaining similar expressions for transitions from higher initial vibrational states. The present model furnishes a broader scheme for the factorization than others.21 It not only relates transitions from different initial states but also relates transitions from the same initial state but to different final states (see eq 17). Consequently any transition amplitude Smn can, in fact, be written as a linear combination of S, and Sloonly. Applying eq 17 (but replacing m by m - 1) and eq 21, one is able to obtain the corresponding factorization of the reactive vibrational probabilities. Thus, squaring the two appropriate expressions and eliminating the cross-product term leads to
The Journal of Physical Chemistry, Vol. 85, No. 26, 198 1 3977
Franck-Condon Model for Collinear Reactive Systems
+
-
+
Next, recalling eq 8 (and assuming the Smn’sare all real), one finds
which is the expression for the factorization of the probabilities.
Numerical Example Below we apply the factorization expressions for the amplitudes. Substitution of the probabilities P,, in eq 21 instead of the corresponding Smn)sby applying eq 9 yields recursion relations for transition probabilities among transitions taking place from the ground state and the first excited state. In order to test the validity of this theory, we applied it to the two isotopic reactions: X + Cl,(Ui = 0, 1) XCl(Uf) + C1 X = H, D (24) +
for which there exist exact state-testate collinear reactive transition probabilities for various e n e r g i e ~ . ~ *In~ the J~ present work only eq 21 is applied. Two new variables are formed which define a point for each final state m
Y m = (Pml/Pmo)l/z (25) z m = [(mkEPm-lo ) / ( ~ E - ~ P ~ o ) I ~ / ~ and consequently eq 21 becomes *Ym = a! - p(*zm)
.=.-(-)
(26)
where
( 2 x ) ’ / , kb 1+x
\
I
Flgure 3. Values of (P,,,r/Pm0)”2as a function of [m(k#,,,-lo)/ (km-,P,,,o)]i~2 for the reactions H Cl,(v, = 0, 1) HCI(v, = m ) CI, m = 1, 2 , ..., 6: (0)E = 0.268 eV; (X) E = 0.50 eV.
p=2-
k:
The signs in eq 26 are not specified, and therefore the relation between the two variables Ymand Zm is well-defined up to a sign. This freedom is used to construct the “best” straight line from the available data. Figure 3 illustrates the results for H + C12 and Figure 4 those for D + C12. Figure 3 gives results for E = 0.268 and 0.5 eV (since the total reactive probability for E = 0.5 eV was somewhat less than 1.0 (-0.9 for n = 0 and -0.8 for n = l ) ,the corresponding points in Figure 3 were obtained after normalization of the reactive probability). The straight line was determined by the least-squares method, using only the points for E = 0.268 eV. The points for E
-
Flgure 4. Values of (P,,,1/Pmo)1/2 as a function of [m(k#,,,..,,)/ (k,,,-lP,,,o)]”2for the reactions D Cl,(v, = 0, 1) DCI(v, = m ) -t CI, m = 1, ..., 7. The polnts are for the energy, E = 0.27 eV.
+
= 0.5 eV were added afterward and are seen to follow the same line. Although p, the gradient of the line, could yield certain information concerning x , in practice it cannot always be uniquely determined. This is because x is very sensitive with respect to small changes in /3 which cannot always be determined accurately enough. For instance, decreasing from 1 to 0.8 will increase x from 1 to 4, and decreasing @ further, to 0.5, will bring x to 14. Under these circumstances, x can at most be estimated to be in the range between 1 and 4. These values are smaller than the ratio of the corresponding frequencies of HC1 and Clz, which is 7 . Thus, it seems as though the transition does not take place between the two original harmonic oscillators. A somewhat different picture is obtained for D + Cl,, for which the results (E = 0.27 eV) are shown in Figure 4. Again the points follow the straight line, but this time x is estimated to be larger, namely, between 5 and 7. Since the ratio between the two frequencies of DCl and C12 is 5, it seems as though the transition takes place between the two undistorted original oscillators. Whereas the gradient yields x , the free parameter a (see eq 27) yields T , the relative shift between the two oscillators. This magnitude is much less sensitive to the numerical inaccuracies and has been found, for the two isotopic reactions, to be equal to 3. This result signifies that the final oscillator should be shifted by an amount equal to 3 times half the width of the (distorted) reagent oscillator for the exact quantum-mechanical calculation of the vibrational distributions to be obtained. The above analysis shows that the two isotopic systems behave differently from the point of view of the dynamics of the process. The transitions from the reagent channel to the product channel in the D Clzcase seem to be more direct (sudden) than in the H C12case, because the FC transitions probably take place between the two original oscillators of Clz and DC1. This is as expected, since the H atom is lighter than the D atom and will therefore adjust itself more gradually once the chlorine atom starts departing. It should be emphasized that the present approach and model are not intended to prove once more that these two reactions, as well as other exothermic reactions, exhibit FC-type processes. This has been done several times in the past, with various degrees of success. In this study we were mainly concerned with the possibility of constructing two linearly dependent variables which could determine without bias and unambiguously the extent to which FCtype processes exist in chemical reactions in general (as well as in other processes). As a byproduct, the approach also yielded an estimate for the two important parameters, the relative shift T , and the ratio between the two oscillator
+ +
3978
J. Phys. Chem. 1981, 85, 3978-3984
frequencies x, which characterize any FC transition.
Conclusion In this work a model was considered which, when ignoring reflection, yields reactive T-matrix elements which are proportional to the Franck-Condon overlap coefficients. These terms were applied to derive factorization expressions for reactive vibrational amplitudes and probabilities. The fiial expressions canbe used to analyze both experimental and numerical results. As an example, we studied the two isotopic systems H(D) + Clz. To do that, we applied the factorization for the amplitudes, rather than that for the probabilities. Here, two variables (one pair for each final vibrational state), which were shown to be
linearly dependent, were constructed. The parameters of the resulting straight lines yielded the ratio between the fundamental frequencies of the two oscillators x and their relative shift T . Whereas x for HClz was smaller than the ratio between the two original frequencies of HC1 and Clz (-3 vs. 7), for DC1, the value of x was similar to the corresponding ratio (which was 5). The relative coordinate shift between the reagent and the product oscillators was found to be equal to 3 in the two cases. Acknowledgment. This research was supported by the Deutsche Forschungsgemeinschaft under Sonderforschungsbereich 91 "Energietransfer bei Atomaren und Molekularen Stossprozessen".
Concentration Dependence of the Diffusion Coefficient of a Dimerizing Protein: Bovine Pancreatic Trypsin Inhibitor Peter R. Wills" and Yannls Georgalls Max-Plenck-Institut fur Molekulare Genetik, Abteilung Wittmann, 1000 West Berlin 33 (Dahlem), West Germany (Received: May 1, 198 1; I n Final Farm: July 28, 198 1)
We examine the theory of the concentrationdependence of the diffusion coefficient of a solution of macromolecules and conclude that an attractive potential, in addition to the usual repulsive hard-core and electrostatic potentials, must be invoked to explain how the diffusion coefficient of spherical protein molecules can decrease with increasing concentration. The case of a narrow square-well potential, modeling dimerization,is considered. The equivalence between the statistical mechanical and thermodynamic descriptions of protein dimerization is demonstrated. It is found that, in the case of the diffusion-controlleddimerization of a protein, the intensity autocorrelation function determined in quasi-elasticlight-scatteringexperiments may be of single-exponentialform rather than the double-exponentialform predicted by the standard theory. These conclusions allow self-consistent interpretation of results obtained by using this technique to investigate diffusion in solutions of the small protein bovine pancreatic trypsin inhibitor.
Introduction Since Batchelor,l Felderhof? and Jones3 presented their resulb pertaining to the hydrodynamic interaction between two spheres suspended in a fluid, it has become possible to build a fairly complete theory of the concentration dependence of the diffusion coefficient of spherical macromolecules."7 The theory is complete in the sense that it uses exact expressions for the particle-pair hydrodynamic interaction tensors and thus allows a result, correct to first order in the concentration, to be obtained, once the nature of any direct pair-interaction potential is specified. For globular proteins, the hard-core potential is minimal and immediately allows use of the results for the hydrodynamic interaction between impermeable particles. Electrostatic interactions are usually modeled in terms of a screened Coulombic potential. Any attractive forces between proteins (hydrogen bonding, hydrophobic interactions, or van der Waals forces) are inherently short-range. When the equilibrium thermodynamic properties of protein solutions are investigated, the effects of these forces are normally reckoned as association constants.8 When first-order concentration effects are being investigated, a dimerization constant is all that need be specified from the thermody-
* Address correspondence to this author a t the following address: Department of Physics, University of Auckland, Private Bag, Auckland. New Zealand.
namic point of view. In the Appendix to this paper, we demonstrate the equivalence between this approach and that used by Batchelorl and Lekkerkerker and co-worke r ~ , who ~ J ~specify instead the depth and the width of a square-well potential. The experiments using the protein bovine pancreatic trypsin inhibitor (BPTI) which we report here are of some intrinsic interest, since they demonstrate that equipment which is now available in many laboratories may be used to determine accurately and quickly the Stokes radius of proteins as small as BPTI, provided they can be obtained at high enough concentration. The molecular weight of BPTI is only 6500, and the crystallographic datal1 indicate that it is a compact globular protein, roughly pear-shaped, with a length of 2.9 nm and a greatest lateral cross-sec(1)G. K. Batchelor, J. Fluid Mech., 52, 245 (1972). (2) B. U. Felderhof, Physica A (Amsterdam),89, 373 (1977). (3) R. B. Jones, Physica A (Amsterdam),92, 545, 557, 571 (1978). (4) G. K. Batchelor, J. Fluid Mech., 74,1 (1976). ( 5 ) B. U. Felderhof, J. Phys. A: Math. Gen., 11, 929 (1978). (6) R. B. Jones, Physica A (Amsterdam),97, 113 (1979). (7) P. R. Wills, J. Phys. A: Math. Gen., in press. (8) P. R. Wills, L. W. Nichol, and R. J. Siezen, Biophys. Chem., 11, 71 (1980). (9) R. Finsy, A. Devriese, and H. Lekkerkerker, J. Chem. Soc., Faraday Trans. 2, 76, 767 (1980). (10)C. Van den Broeck, F. Lostak, and H. N. W. Lekkerkerker, J. Chem. Phys., 74, 2006 (1981). (11) R. Huber, D. Kukla, A. Riihlmann, 0. Epp, and H. Formanek, Naturwissenschaften, 57, 389 (1970).
0022-3654/81/2085-3978$01.25/00 1981 American Chemical Society
