J . Phys. Chem. 1987, 91, 5314-5324
5314
evidence that, in transition-metal-ion/moleculereactions, an electrostatically bound complex exists for some time before bond cleavages occur.Is One possible scenario is that the ion "parks" at some relatively large distance from the molecule to form the complex and the chemistry that follows draws the ion closer to atoms in the molecule. Another possibility is that the ion and molecule are in close proximity in the complex and motion of groups of atoms within the molecule must occur for the chemistry to begin; that is, the events are sequential in time but not necessarily corresponding to radically different separations. While it may be that the repulsive terms are too small in these calculations, it appears that there is little to stop the ion from getting close to the molecule, forming the initial complex, suggesting the latter scenario. This may not be the case for polar and/or unsaturated molecules in which a substantial charge distribution may exist throughout the structure.
first-row transition-metal ions, these calculations suggest that a complex in which an ion is in close proximity to the interior C-C bond is more strongly bound than that for a terminal C-C bond. This is easily visualized in the context of the bond polarizability description utilized here, which suggests that ion/induced dipole interactions involving C-C bonds are more extensive when the metal ion lies along a C-C bond axis than when positioned atop such a bond. Only when the metal ion is close to the interior C-C bond can it interact with two C-C bonds, as opposed to interaction with only one C-C bond when it is close to a terminal C-C bond. This simple approach to the modeling of ion/molecule complexes suggests that the first step in these reactionscomplexation-may be a key factor in determining which bonds are preferentially attacked and, hence, the final product distributions. Work is under way to refine the model and extend this work to larger alkanes, polar compounds, and metal ions with asymmetric charge distributions.
Conclusions By use of a simple model, preferred configurations of electrostatically bound metal ion/butane complexes are identified. This work supports the concept that ion/molecule interactions can be substantial, not only for polar molecules but for nonpolar, polarizable species as well. Concerning the chemistry of butane with
The Addition and Dissociation Reaction H
Acknowledgment. Partial support for this project from Professor Paul Hunt, Director of Academic Computing at Michigan State University is acknowledged. Also, we thank J. F. Harrison and K. C. Hunt for helpful discussions. Registry No. CO', 16610-75-6; n-C4H,,,, 106-97-8.
+ CO C HCO. 1. Theoretical RRKM Studies
Albert F. Wagner* and Joel M. Bowman' Theoretical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (Received: December 29, 1986)
RRKM calculations of the thermal addition and dissociation rate constants are carried out on the Harding surface for HCO. Structures, frequencies, and energetics at the stationary points on the surface are presented for HCO and DCO. Due to the relatively weak H-CO bond (bond energy of about 16 kcal/mol) and the few degrees of freedom in HCO, the spacing between the vibrational states of the metastable HCO* is quite large. This motivates a form of RRKM theory in which metastable HCO* is explicitly considered to exist only at isolated vibrational resonance energies and all addition and dissociation dynamics is controlled by these isolated resonance states. The rate constant calculations focus on the implications of this isolated resonance model. Several experiments to test the model are indicated. Among the major conclusions of the study are (1) a standard treatment of tunneling cannot be included in the calculations, (2) explicit summation over total angular momentum must be included in the calculations, (3) interesting isotope effects in addition are predicted in the low-pressure limit of the rate constants, and (4) measurable recrossing effects are predicted in the approach to the high-pressure limit of the rate constants. In the following paper, the isolated resonance RRKM calculations are directly and generally favorably compared to the existing low-pressure measurements for both thermal addition and dissociation.
I. Introduction The collision dynamics of H with CO has been studied over the years in three different energy regimes. In the highest energy regime, hot atom sources for H atoms have been used to probe inelastic dynamics at energies up to several electronvolts. Both e~perimentall-~ and interpretative theoretical studies& have been carried out. In an intermediate energy regime, thermal addition and dissociation of HCO in a buffer gas M
+
H co HCO* 2 HCO (1) have been extensively studied e~perimentally.~-'* The rate constants for these processes are of interest in combustion chemistry. However, essentially all the experimental studies at combustion temperatures (1200-2500 K) are indirect of the dissociation rate constant, and there is no agreement on the rate constant to within at least 1 order of magnitude. There are three experimental studies"'-'2 of the addition process near room temperature, all of which indicate a relatively low rate constant. With 'Permanent address: Department of Chemistry, Emory University, Atlanta, GA 30322.
0022-3654/87/2091-5314$01.50/0
the exception of one model study,I3 there have been no theoretical studies of these rate constants. In the lowest energy regime, there (1) Wood, C. F.; Flynn, G . W.; Weston, Jr., R. E. J . Cbem. Phys. 1982, 77, 4776. (2) Wight, C. A.; Leone, S. R. J . Chem. Pbys.'1983, 78, 4875. (3) Wight, C. A,; Leone, S. R. J . Chem. Pbys. 1983, 79, 4823. (4) Gieger, L. C.; Schatz, G. C. J . Phys. Chem. 1984, 88, 214. (5) Gieger, L. C.; Schatz, G. C.; Garrett, B. C. In Resonances in Electron-Molecule Scattering, van der Waals Complexes, and Reactiue Chemical Dynamics; Truhlar, D. G:, Ed.; American Chemical Society: Washington, DC, 1984; Chapter 22. (6) Gieger, L. C.; Schatz, G. C.; Harding, L. B. Chem. Phys. Lett. 1985, 114, 520. (7) Baulch, D. L.; Drysdale, D. D.; Duxbury, J.; Grant, S . J. Evaluated Kinetic Data for High Temperature Reactions; Butterworths: London, 1976; Vol. 3, p 327 ff. (8) Warnatz, J. In Combustion Chemistry; Gardiner, W. C . , Ed.; Springer-Verlag: New York, 1984; Chapter 5 . (9) Hucknall, D. J. Chemistry of Hydrocarbon Combustion; Chapman and Hall: New York, 1985; p 304 ff. (10) Hikida, T.; Eyre, J. A.; Dorfman, L. M . J . Chem. Phys. 1971, 54, 3422. ( 1 1) Wang, H. Y.; Eyre, J. A.; Dorfman, L. M . J . Chem. Phys. 1973,59,
5199.
0 1987 American Chemical Society
H
+ C O F= H C O Reaction
have been experimentalJ4and theoreticall5-I9studies of the inelastic single-collision dynamics at energies below and just above the barrier to addition (about 2.0 kcal/mol). These studies are of interest in astrophysical applications. Recently, a new global potential energy surface of H C O has become available. This surface was produced by HardingZ0with ab initio electronic structure methods and has been tested in a variety of ways. The major tests20-2'have been by comparison with the recently measured vibrational spectra of H C O and DCO by photodetachment of HCO- and DCO-.22 The comparisons are quite favorable (less than a 20-cm-' average error for all the bound vibrational states in H C O or DCO). This new surface has also been used in calculations of both the high-energy and low-energy inelastic dynamics mentioned above. Comparison to the hot atom experiments via classical trajectory simulations6 on the potential energy surface have produced overall favorable agreement, but some disagreements remain. Low-energy coupled channel inelastic calculation^^^-^^ have recently been examined on the potential energy surface, and a comparison of resonance energies to experiment was made.I9 In this paper and the paper immediately following23(hereafter referred to as paper 2), calculations of the thermal addition and dissociation rate constants are performed on this new potential energy surface. RRKM theory is the method of calculation, and this paper focuses on the special features and modifications of this theory as applied to HCO. Paper 2 reports recent direct dissociation rate constant measurements and the comparison of the theory to these and all other available thermal dissociation or addition experiments. The paper is organized as follows. In the next section, the characteristics of the potential energy surface pertinent to thermal association and dissociation are briefly described. In section 111, the modifications of RRKM theory necessary to describe the dynamics of H C O are discussed. The sparse set of molecular vibrational states present in H C O motivates an unconventional form of RRKM theory. In section IV, rate constant calculations are carried out which illustrate the special characteristics of the H C O thermal dynamics as predicted by the modified RRKM theory. These calculations suggest experiments that would be able to test the approximation made in the theory. Section V is a summary. This paper and the following paper 2 are the first in a series of thermal dynamics studies on HCO. Subsequent studies will include quantum and quasi-classical trajectory calculations and will examine the importance of variational and anharmonic effects and refine the generally successful comparison to experiment these first calculations suggest. 11. Potential Surface The details of the calculation of the potential energy surface are available in ref 20. In brief, for approximately 2000 geometries, the potential energy was determined by an ab initio restricted Hartree-Fock calculation followed by a configuration interaction calculation consisting of all single and double exci(12) Ahumada, J. J.; Michael, J. V.; Osborne, D. T. J. Chem. Phys. 1972, 57, 3736. (13) Bowman, J. M.; Lee, K.-T.; Romanowski, H.; Harding, L. B. In Resonances in Electron-Molecule Scattering, van der Waals Complexes, and Reactive Chemical Dynamics; Truhlar, D. G., Ed.; American Chemical Society: Washington, DC, 1984; Chapter 4. (14) Caracciolo, G.;Ellis, T. H.; Este, G. 0.; Ruffolo, A,; Scoles, G.; Valbura, V. Astrophys. J . 1979, 229, 451. (15) Chu, S.; Delgarno, A. Proc. R . SOC.London, A 1975, 342, 194. (16) Green, S . ; Thaddeus, P. Astrophys. J . 1976, 205, 766. (17) Romanowski, H.; Lee, K.-T.; Bowman, J. M.; Harding, L. B. J . Chem. Phys. 1986, 84, 4888. (18) Lee, K.-T.; Bowman, J. M. J . Chem. Phys. 1987, 86, 215. (19) Lee, K.-T.; Bowman, J. M. J . Chem. Phys. 1986, 85, 6225. (20) Bowman, J. M.; Bittman, J. S.; Harding, L. B. J . Chem. Phys. 1986, 85, 911. (21) Christoffel, K. M.; Bowman, J. M.; Bittman, J. S. Chem. Phys. Lett. 1987, 133, 525. (22) Murray, K. K.; Miller, T.M.; Leopold, D. G.; Lineberger, W. C. J . Chem. Phys. 1986, 84, 2520. (23) Timonen, R. S.; Ratajczak, E.; Gutman, D.; Wagner, A. F. J . Phys. Chem., following paper in this issue.
The Journal of Physical Chemistry, Vol. 91, No. 20, 1987
5315
TABLE I: Equilibrium Geometries, Principal Moments of Inertia, Harmonic Frequencies, Zero-Point Energies, and Relative Enthalpies CO, HCO', and HCO
for H
+
property
HCO H
DCO
+ CO
Rco. A I , amu A2 w, cm-] ZPE, kcal/mol
1.150 9.07 2173 3.1 1
HCO* Rco, A RCHi 8, OHCO,
1.154 1.850 117.2
deg
I,, amu A2 I,, amu A2 I,, amu A2 wl(CO), cm-'
w2(HCO),cm-I w,(CH, imaginary), cm-' ZPE, kcal/mol
2.01 11.96 13.97 2120 400 589 3.60
3.00 15.47 18.47 2119 324 423 3.49 1.195 1.124 124.2
0.71 11.61 12.32 1903 1145 2748 8.29 2.1 -14.2
1.18 13.50 14.67 1861 897 2054 6.88 2.0 -15.6
tations and a correction for quadruple excitations. The basis set used in the calculation was double-{ with polarization functions. In a novel scheme, a three-dimensional spline function was fit to these 2000 computed energies to form a global HCO surface. This global surface had an activation energy for addition (at 0 K) of 5.8 kcal/mol and overall addition enthalpy (at 0 K) of 18.1 kcal/mol (for H C O HCO). As discussed in paper 2, the only empirical information on the addition barrier would suggest an activation energy of about 2.0 kcal/mol (at room temperature) and the consensus value of the overall addition enthalpy would be 14.2 f 1.0 kcal/mol. To remove these discrepancies, the ab initio surface was empirically scaled. The details of the scaling function are in ref 20. The first result of the scaling was to lower the H C O well depth to agree with the overall addition enthalpy but to negligibly change the geometry and frequencies of HCO. The second result was to substantially lower and broaden the addition barrier but to negligibly change the geometries and frequencies of the addition transition state. The resulting addition activation energy (at 0 K), Le., the zero-point adiabatic barrier, was 2.1 kcal/mol. All the calculations in this and the subsequent paper are based on this empirically scaled surface. The RRKM calculations require a characterization of only the extrema of the surface, namely, the reactants, H and C O for addition and H C O for dissociation, and the addition (or dissociation) transition state, HCO*. That characterization is in the form of the structure (in terms of moments of inertia), the harmonic vibrational frequencies, and the energetics. Table I provides all the necessary information for both H C O and DCO. The information is obtained by a local least-squares fit to a Simons-Parr-Finlan type power series expansion of the potential. Equilibrium geometries, moments of inertia, and harmonic vibrational frequencies were then derived by using techniques described previously. The HCO information was previously reported in ref 20, but full DCO information is reported here for the first time. As mentioned in the Introduction, the accuracy of the H C O well region of this surface has been confirmed by comparison to photodetachment spectra. The comparison to thermal experiments in paper 2 will test the accuracy of this surface at energies com-
+
-
5316 The Journal of Physical Chemistry, Vol. 91, No. 20, 1987
Wagner and Bowman
parable to the addition barrier height. In that comparison, slight variations in the addition barrier height will be explored. A small correction to the addition barrier height is then suggested.
7000.0 6000.0
111. RRKM Theory The special nature of the H C O molecule motivates an unconventional form of RRKM theory based on isolated resonances. The basic nature of this form, followed by the details of the calculation, is presented below. A . Isolated Resonance R R K M Theory. RRKM theory can be used to construct a temperature- and pressure-dependent association rate constant of the form24
t==
-E
5000.0
4000.0 F‘.
E
3000.0
w 2000.0
1000.0
where
................... ................... o.o ................... ...................
and
-1000.0
1E,=, c P(Ev)l/hP(E)
!
1.2
E-E,
WE) =
iP
(4)
In eq 2-4, k,[M] is the effective stabilization rate constant as a function of the pressure (P)-dependent concentration [MI of the buffer gas M, cP(E,,) is the sum of vibrational states open at the transition state at energy E - Eo, Eo is the threshold for dissociation including zero-point effects (Le., M(0 K)H+C+.H-.CO AH(0 K)H+CO-HCO in Table I), p(E) is the density of H C O molecular vibrational states at total energy E above the zero-point energy at the bottom of the well, Q( T) is the partition function T) is the partition at temperature T for reactants H + CO, QIext-rot( function at the transition state for external rotation, fl is the zero-point corrected barrier to addition (i.e., M(0 K)H+C+.H...CO in Table I), h is Planck’s constant, and k B is the Boltzmann constant. This formula ignores complications of internal rotations, total angular momentum conservation, active or adiabatic treatment of azimuthal external rotations, and variational effects. Such considerations will be discussed later. This formula for the association rate constant can be converted to the dissociation rate constant via the equilibrium constant. The special nature of R R K M theory when applied to the dynamics of H C O primarily involves the treatment of p ( E ) . Generally, p(E) is calculated in the harmonic approximation using the harmonic vibrational frequencies of the molecule, in this case HCO. Generally, p ( E ) is an exponentially increasing function of energy in the energy range of interest ( E , and above), and for most reactions studied by R R K M theory, p ( E ) will correspond to the presence of many states per cm-I. The presence of so many states in a small interval of energy is the basis of the randomization approximation in RRKM theory; Le., all states at the same total energy equally participate in the unimolecular dynamics no matter what states were initially populated. (These “states” are of course scattering resonances.) Thus, to agree with experiment it is sufficient for RRKM theory to be accurate on the average. (Of course, it may also be accurate for every respnance state, as well.) In HCO, the density of vibrational states is so small that p(E) is best considered a series of isolated delta functions centered on resonance energies over the whole range of collision energies of interest. In other words, for any reasonable graining of the energy spectrum, most energy intervals contain no vibrational states, and those few that do contain only one state (i.e., there are no accidental degeneracies). This is illustrated in Figure 1, where the harmonic vibrational energy levels of HCO up to 7000 cm-’ above the H + C O asymptote are indicated relative to a schematic of the zero-point corrected potential energy profile along the reaction path. The four states below the addition barrier are indicated with dotted lines because they can only directly participate in the
+
(24) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; WileyInterscience: New York, 1972; Chapter 4 .
I
1.5
I
I
I
I
1.8
2.1
2.4
2.7
.O
Rtl-co (A0) Figure 1. The solid curved line is a schematic of the zero-point corrected potential energy vs. progress along the reaction path as measured by the separation of H from the C in CO. The solid horizontal lines are the energies of the HCO vibrational states in the harmonic approximation
that lie above the potential energy curve. The dashed lines are those molecular states in the harmonic approximation that could dissociate only via tunneling. dynamics by tunneling. The figure clearly shows hundreds of wavenumber spacings even up to the highest energies. This feature is caused by the combination of a low bond energy ( E , = 16.0 kcal/mol) and a light atom (H) involved in two of the three modes of vibration. For all molecules, p(E) is a discrete series of delta functions at low enough energies where the vibrational spectrum displays separated vibrational lines. However, unlike the case in HCO, p(E) for most molecules loses this characteristic at energies relevant to dissociation and addition because the vibrational resonance states are so close together in energy their resonance widths overlap. Thus, p ( E ) for many molecules consists not of a discrete series of isolated resonances but a smooth continuous function of overlapping resonances. The typical approach in carrying out a practical RRKM calculation for H C O would be to approximate p ( E ) by some smooth function. However, this approach is arbitrary for there is no physical basis for the smoothing. The most obvious way to obtain a smooth representation of p ( E ) would be to use the semiclassical Whitten-Rabinovitch a p p r o x i m a t i ~ n . ~This ~ approximation was originally derived by fitting direct count values for p ( E ) for “typical” molecules with a functional form in reduced energy space. Although designed for a rapid approximate calculation of p ( E ) , the method produces perforce a continuous representation of p ( E ) . An alternative method would be a strictly numerical smoothing method such as rolling averages. In this procedure, a new value of p ( E ) is constructed from the average of the old values of p ( E ) in an input energy window centered about E. After a new p ( E ) has been created for the whole energy spectrum, the process is repeated as many times as required to make p ( E ) everywhere larger than some input value. [This rolling average procedure is relatively insensitive to the width of the energy window and to the minimum value of p ( E ) (as long as it is relatively small).] The method is, in effect, a mathematical way of broadening the true isolated resonances until they significantly overlap. The result for H C O is seen in Figure 2, where the WhittenRabinovitch representation and rolling average representation (energy window of 15 cm-’ and minimum value of states/ cm-*)are compared to the original isolated resonance form of p ( E ) .
-
(25) This approximation is reviewed in ref 24, Chapter 5. (26) Forst, W. Theory of Unimolecular Reactions; Academic: New York, 1973; Chapter 5 .
H
+ CO
0.0
The Journal of Physical Chemistry, Vol. 91, No. 20, 1987 5317
H C O Reaction
mo.0
wo.0
8.0
2100.0 20b(
E (an-’)
Figure 2. Density of HCO vibrational states p(E) in the harmonic approximation vs. energy E above the H + CO asymptote. The stick spectrum is a representation of the delta function density of resonance states. The dashed line is the Whitten-Rabinovitch approximation. The solid line is a rolling average approximation to the density of states
described in the text. The figure shows that the Whitten-Rabinovitch p ( E ) is an approximate average of the highly oscillatory rolling average p ( E ) which in turn is an extremely broadened form of the stick spectrum of the isolated resonances in the original p ( E ) . Note that the energy widths of the rolling average oscillations and the larger spacings between the isolated resonances are of the order of 1 kcal/mol or 500 K in kBT. Thus, the selection of different smooth representations of p ( E ) could influence both rate constant magnitude and temperature dependence (Le., activation energy) at lower temperatures. A more compelling approach in carrying out a practical RRKM calculation for H C O is to accept the isolated resonance nature of p(E) and incorporate this feature in the RRKM formulas. This is simple to do: k(E) in eq 4 is presumed to be nonzero only at the isolated resonance energies, and the integral in eq 2 is replaced by a summation over these energies
where AE, is the resonance width at the ith resonance E,. While simple in principle to implement, this approach embodies the significant assumption that all the dynamics in addition or dissociation takes place only at the discrete energies where p ( E ) is nonzero. In other words, the dynamics of H C O is exclusively controlled by isolated resonances. This assumption of isolated resonance control implies two major changes in the standard interpretation of RRKM theory. The first concerns the sum of states C P ( E , ) , which is a measure of the amount of phase space available at the top of the barrier to addition or dissociation. This is a continuous function of E . However, in an isolated resonance form of RRKM theory, only resonance width slices of this continuous function at the discrete resonant energies participate in the dynamics. The unused portions of this function still represent accessible regions of phase space at the barrier to addition and dissociation but regions that cannot be correlated with a resonance of the molecule. The nonresonant scattering is strictly a quantum mechanical effect because of the discrete nature of the resonance energy spectrum. By contrast, at all energies (above the barrier), some classical trajectories will form long-lived collision complexes. However, without the imposition of some quantization principle (either exactly through quantum mechanics or indirectly through semiclassical arguments), the discrete quantum resonance cannot be described classically. Other trajectories will cross the transition state, enter the complex region of space but not form a long-lived complex, and instead promptly recross the transition state to form the reactants. Thus, these trajectories could be labeled recrossing trajectories in analogy with the use of that term in bimolecular exchange reactions, where it originated. The term “recrossing” can also be applied to the quantum scattering calculations in the same sense as it applies to the classical ones but with an important
difference. In the quantum calculations there is complete recrossing at energies which do not correspond to resonance energies. That is, like classical recrossing, although the quantum scattering wave functions have some amplitude in the region of the complex, a resonance state does not form. It is in this sense that ”recrossing” will also be used to describe the quantum scattering at energies off resonance. The second change in the standard interpretation of RRKM theory has to do with the randomization approximation. Generally, the statistical nature of RRKM theory stems from the assumption that, no matter how the metastable molecule acquires a given total energy and angular momentum, the molecule undergoes complete intermolecular relaxation to all accessible molecular states on a time scale faster than any other process. In other words, the total energy is randomized among all possible molecular states within the constraints of total angular momentum. In most applications of RRKM theory, there are many molecular states at the same energy (within the width of the energy grain size of the practical calculation) and the meaning of this approximation is clear. In HCO, there is only one molecular state and in that sense there is no randomization possible. The randomization approximation is applied for HCO on the more detailed level of phase space: all the phase space represented by each metastable HCO* state is fully sampled by the collision dynamics on a time scale faster than any other process. This more detailed application of the randomization approximation makes RRKM theory less likely to be accurate for HCO, but the comparisons to experiment presented in paper 2 will show the theory to be quite useful in representing available experimental results. There is one caveat concerning the isolated resonance model for H C O that must be discussed. The large separation between resonances is a feature of the vibrational states for each total angular momentum characterized by the quantum number J . The results in Figures 1 and 2 are for J = 0. However, H C O really has uibrational-rotational states where the rotational component depends not only on the conserved quantum number J but on the unconserved quantum number K characterizing the projection of the total angular momentum on the body-fixed axis. Thus for each value of J , the vibrational-rotational spectrum of states would be characterized by the vibrational level spacings of Figure 1 augmented for each vibrational level by the tumbling rotational levels characterized by K . If H C O is approximated as a prolate symmetric top with the larger moment of inertia approximated by the geometric mean of the two larger moments of inertia for HCO, then, given Table I, the rotational constant for this Kdependent rotational energy is about 20 cm-’. Since the rotational energy goes as p,the typical spacing between vibrational-rotational levels for a fixed J would be on the order of tens of cm-I at higher energies, much smaller than that displayed for vibrational spacings alone in Figure 1 but still significantly larger than most molecules. The significance of this increased density of molecular states depends on whether the tumbling degree of freedom characterized by K behaves adiabatically or actively during the course of a unimolecular dissociation. If active, tumbling energy can readily flow into the vibrational degree of freedom, and vice versa. Then the increased density of states is quite relevant, and the likelihood of an isolated resonance model being accurate decreases because the molecular resonance states are much closer together. On the other hand, if tumbling is adiabatic, the K quantum number is conserved and it is the density of vibrational resonance states for each value of J and K (cf. Figure 1) that is relevant. In this case, the isolated resonance model should be appropriate for the reasons given above. The actual behavior of tumbling will vary somewhere between these two extremes, with deviation from adiabaticity controlled by vibration-rotation interaction through centrifugal stretching or Coriolis c ~ u p l i n g The . ~ ~assumption ~~~ that K is a good quantum number is currently being investigated in coupled channel scattering calculations; however, in the absence of evidence to the contrary, we will make the simpler assumption that it is a good quantum number.*’ (It should be noted that in a classical tra-
5318
The Journal of Physical Chemistry, Vol. 91, No. 20, 1987
jectory study of complex formation in H + O2K appears under some conditions to be active, at least c l a ~ s i c a l l y . ~ ~ ) E. Detailed Specifcation of the Calculation. The remaining aspects of the RRKM calculations can now be specified. All the densities of states and sums of states were determined by direct count, using the Beyer-Swinehart algorithm.29 Summation over total angular momentum J is explicitly included. (This summation is not included in eq 2 but is discussed in more detail below.) The external tumbling rotation characterized by K is classified as adiabatic. Tunneling through the H C O addition barrier is not included. The stabilization rate constant k, is represented by the Lennard-Jones gas kinetic rate constant times an efficiency factor for stabilization. The Lennard-Jones parameters are taken from the tabulation of ref 30 with the values for acetylene used for HCO*. The efficiency factor, &, is determined in a way to empirically mimic a master equation solution according to ref 31. p, requires as input the average energy ( AI?),,,transferred between buffer gas and metastable complex per up and down collisions. The value selected will be discussed shortly. In these calculations, the rigorous explicit summation over the K quantum number was approximated by a thermal average. To do this and the explicit J summation, H C O was assumed to be a symmetric top. This is a reasonable assumption because, from Table I, the lowest moment of inertia is less than 10%of the two larger moments which in turn are within 10% of each other. The eigenvalues eJK for a symmetric top have the well-known form
+
EJK = A J ( J 1 ) -k ( E - A ) K 2 (6) where A = h2/8rrZ, and E = h2/8aZ, with Z, being the larger, and Z, the smaller, of the two distinct moments of inertia of the in eq 2 has the top. In the symmetric top approximation, Q*exr-rot form m
Qtext-rot =
J
( 2 J + l)e-AJ(J+l)/kBT[ 2 e-(B-A)K2/kBT] (7) K=-J
J=O
The entire J dependence of the term in brackets resides in the limit on the summation. If Z, is sufficiently smaller than I,, then E - A becomes large enough so that the terms in the sum for the limits on K become very small and contribute negligibly to the sum. Under such circumstances, the fJ limits on the summation can be replaced by f m with negligible loss of accuracy. Qtext-rot can than be factored into the J summation in eq 7 times the K summation extended to m which is in fact the partition function for a one-dimensional rotor. The full J and K dependence in the rate constant k( T,P) can in the equation by its be included in eq 2 by substituting Q*ext-rot exact form in eq 7 and recognizing that bothf(E) and k ( E ) in the equation have a J and K dependence
-
f(E)-ff(E,J,K) k ( E ) k(E,J,K) (8) that comes from the fact that both the sum of HCOT states and the density of H C O states depend on the effective potential which contains the J and K symmetric top terms.24 Note that the K dependence of the effective potential is a consequence not of the conservation of total angular momentum (which determines the J dependence) but rather of the approximation discussed above that tumbling motion is adiabatic (Le., the value of K is unchanged during unimolecular dissociation). In its full form, k ( T , P ) will include a summation over J and K and an integral over E . Often in RRKM calculations of k( T,P), the J and K summations are drastically simplified by an approximation in which the (27) Quantum scattering calculations for H + CO are currently in progress and will be analyzed for resonances and for changes in K involving resonances from that part of the scattering that samples the HCO well. (28) Brown, N. J.; Miller, J. A. J . Phys. Chem. 1982, 86, 772. (29) Asholtz, D. C.; Troe, J.; Wieters, W. J . Chem. Phys. 1979, 70, 5107. (30) Hippler, H.; Troe, J.; Wendelken, H. J. J . Cbem. Phys. 1983, 78, 6709. (31) Troe, J. J . Chem. Phys. 1977,66,4745. The efficiency factor derived here is for low-pressure thermal dissociation, the reverse of addition. The ratio between the addition and dissociation nonequilibrium rate constants from the master equation has been shown to be the equilibrium constant (Troe, J. Annu. Rev. Phys. Chem. 1978, 29, 223) which implies the derived efficiency factor applies to low-pressure addition as used here.
Wagner and Bowman J and K dependences of f ( E , J , K ) and k ( E , J , K )are replaced by an average value:24 f(E,J,K) - f ( E , ( J ) , ( K ) ) k(E,J,K) k ( E , ( J ) , ( K ) ) (9) The average is based on the fact that classically the average energy in the metastable molecule at temperature Tis 3/2kBTfor the three external rotational degrees of freedom. To the same degree that Q*ext-ro, can be factored into K- and J-dependent factors, the 3/2kBT can be assigned as '/2kBT for the K-dependent degree of freedom and kBT for the J (and implicitly JJdependent degrees of freedom. If we equate this average energy with the J- and Kdependent terms in the symmetric top eigenvalue expression (eq 6), average values ( J ) and (K) can be determined that are temperature dependent. (The moments of inertia to use in eq 6 are those for H--C0.24) With approximation eq 9, the J and K summations become entirely contained in Q$ext-rot where they can be evaluated exactly or with highly accurate closed-form classical formulas. In most of the calculations reported below, the J summation was not approximated. For HCO, this approximation actually breaks down for reasons that will be discussed through comparisons between the rigorous summation and the ( J ) approximation. However, the (K) approximation is used throughout for reasons of economy. For HCO, this approximation may introduce small errors that will be discussed later. Given the basic form of the calculations, the input that must be specified is the bath gas M, the value of (AE),,, as a function of temperature, the resonance widths AE,, and the structures, frequencies, and energies of H CO, HCOI, and HCO. In the calculations in this paper, the bath gas is He and the structures and frequencies are taken from Table I. The resonance widths AE,are set to 1 cm-I; however, as will be shown shortly, the actual value of this parameter has no influence on the rate constant for almost all the calculations shown here. (AE),,, is considered independent of temperature, as suggested by the most recent measurements of this d e p e n d e n ~ e . The ~ ~ remaining parameters will be varied with the standard set of values being 75 cm-I for (AE),,, and 15.9 and 1.9 kcal/mol for E, and fl. These standard values are similar, but not identical, to those shown in paper 2 to give good agreement to available experimental information on the He-moderated addition and dissociation of HCO. The major difference is that the energetic parameters in paper 2 are several tenths of a kcal/mol lower than the standard values which in turn are a few tenths of a kcal/mol different from the results shown in Table I.
-
+
IV. Rate Constant Calculations and Discussion The RRKM theory of isolated resonances as described in the previous section is used in paper 2 to compare to both new and previously published measurements. The agreement is generally favorable. In this section, calculations are presented which focus on six aspects of the theory that are relatively independent of the reaction studied: (a) tunneling, (b) total angular momentum summation, (c) energy parameter dependence in the low-pressure limit, (d) isotope effects, (e) recrossing, and (f) stabilization rate constants for molecules with isolated resonances. In the course of discussing these results, several possible experimental tests of the theory are proposed. A . Tunneling. A standard treatment of tunneling through the addition barrier must not be included in the RRKM calculations if the calculated rate constants are to have a qualitatively reasonable pressure dependence. This can readily be seen by rearranging eq 2 for k ( T , P ) into a form that anticipates the lowpressure limit:
(32) H q n i n n n . Xf.; Hippler, H.; Troe, J. J . Cbem. Phys. 1984, 80, 1853
H
+ CO
The Journal of Physical Chemistry, Vol. 91, No. 20, 1987 5319
H C O Reaction
1.2
Y
H + CO -+ HCO
0.81
0.4
,:I‘
-I
I
--.._
I, ...........Tu”EuNc!..~r.=..~~ooo~....
vll” ....
0.0
I
0.0
200.0
I
I
I
400.0
600.0
1
I
,
800.0 1000.0 1200.0 U00.0 1600.0
P(t0rr) Figure 3. Calculated addition rate constant (in cm’/(molecule s)) divided by the pressure (in Torr) of H e buffer gas vs. the pressure of H e at room temperature. Calculations with and - - -) and without (-) tunneling a r e represented with the total average energy change ( A E ) , , , shown. (-e
This rearrangement uses the definitions off(E) and k ( E ) in eq 3 and 4. Elowis the minimum energy, as measured from the bottom of the well, that a metastable molecule can have, Le., Elow = Eo without tunneling or -AH(O K)H+C04HC0 with tunneling. Equation 10 is still in the limit of a continuous density of states. In the case of isolated resonance, the delta function form of p ( E ) will convert the integral into a summation. The low-pressure limit is reached when k,[M] becomes much less than k ( E ) over the range of E strongly weighted in the integral (or summation). Then the integral (or summation) will be independent of [MI and the rate constant as a whole will be linear in the pressure-dependent [MI factor outside the integral (or summation). The Boltzmann factor e-(E-Eo)lknT weighs most heavily the low end of the E range. If p ( E ) is a very weak function of energy, then the Boltzmann factor alone dominates and, to be in the low-pressure limit, k,[M] must be less than k ( E ) for the lowest energies. In an isolated resonance model, p ( E ) is by definition independent of E except for resonance locations because every isolated state receives a unit weight in eq 10. That is to say that the conversion of the integral of eq 10 to a summation over isolated resonances includes the transformation p(E) d E
-
p(E,)AE, = 1
(11)
(It is for this reason that the specific value of AEl is of no consequence in the calculations of rate constants near the low-pressure limit.) For HCO, even in the continuous approximations to p ( E ) , the energy dependence is very slight. For example in Figure 2, p(E) in the Whitten-Rabinovitch approximation changes by only a factor of 2 over a 7000-cm-I change in E . Thus for HCO, the lowest energy states determine the pressure at which the lowpressure limit is attained. The tunneling states are always the lowest energy states (see Figure 1) and hence count most heavily in H C O in determining if the low-pressure limit is reached. For these energies, k ( E ) has the small size expected of a classically forbidden process. Thus, to be in the low-pressure limit for HCO, k,[M] must also be no larger than the size of a classically forbidden process. However, there is no physical reason for this to be so. The stabilization of a highly energetic metastable molecule by multiple collisions occurs by classically allowed inelastic processes. Both measurem e n t ~ ~and ~ ,classical ~ ~ - ~trajectory ~ simulation^^^^^^ agree that stabilization rates are typical of classically allowed processes. Thus, tunneling states (with unit weight for p ( E ) ) must be excluded from the calculation. (33) Dove, J. E.; Hippler, H.; Troe, J. J . Chem. Phys. 1985, 82, 1907. (34) Hippler, H.; Troe, J.; Wendelken, H. J. J . Chem. Phys. 1983, 78, 67 1a. (35) Hippler, H.; Lindemann, L.; Troe, J. J . Chem. Phys. 1985, 83, 3906. (36) Brown, N. J.; Miller, J. A. J . Chem. Phys. 1980, 80,5568. (37) Gallucci, C. R.; Schatz, G . C. J . Phys. Chem. 1982, 86, 2352.
To illustrate the importance of the exclusion of tunneling, in Figure 3 the calculated room-temperature addition rate constant divided by pressure is plotted as a function of He pressure for three different calculations. For all three cases, the calculations are normalized at 50 Torr to highlight their relative changes with pressure. The standard calculation excludes tunneling and is very weakly dependent on pressure because it is not quite at the lowpressure limit. (This calculation compares quite favorably to experiment as can be seen in paper 2.) The second calculation includes the four tunneling states in Figure 2 and calculates the sum of states for all states via the Eckart tunneling formula as first suggested by Miller.38 The figure clearly shows that including tunneling dramatically increases the pressure dependence of the results. The third calculation shows what reduction in k,[M] is required in order for a tunneling calculation to reproduce the standard calculation’s insensitivity to pressure. As discussed earlier, the size of k, is set by the gas kinetic collision rate (which Only when this latter is only geometry dependent) and by (AE),,. parameter is reduced from -75 to -0.0002 cm-l does pressure insensitivity result in the presence of tunneling. This value for (AE)totis many orders of magnitude lower than any measured value for a highly energetic molecule and is only consistent with a classically forbidden process. The requirement that tunneling be excluded from the RRKM calculations is not consistent with RRKM theory. As first outlined by Miller3*and as is consistent with numerous theoretical studies of closely related transition-state t h e ~ r y ,tunneling ~ ~ . ~ ~ naturally fits into RRKM theory by allowing the probability of a transition state being open (Le., P(E,) in the sum of states CP(E,)) to have a continuum of values from 0 to 1 according to the quantum mechanical probability of traveling over the addition barrier. In a strictly classical picture, P(E,) is either 0 or 1. At energies well above the barrier, classical and quantum values of P(E,) are both unity. At energies below the barrier, classical values are zero while quantum values are low but nonzero. Therefore, the exclusion of tunneling should be thought of as an additional assumption beyond RRKM theory. One of us (Bowman) has recently p r o p o ~ e ad theory ~ ~ ~ ~for~ thermal addition and dissociation that is derived directly from quantum scattering theory under the condition that the reaction is dominated by isolated resonances. This theory is more rigorous than RRKM theory and can offer a tentative explanation of why tunneling should not be included in the RRKM calculations for HCO. That explanation is now presented below. Calculations on H C O with this more rigorous theory are in progress2’ and, when completed, will allow direct analysis of this tunneling question. As described in detail el~ewhere,~l.~* the RRKM sum of states CP(E,) and density of states p ( E ) can be replaced with more rigorous expressions that depend only on the S matrix determined in an exact quantum scattering calculation. (For notational simplicity, angular momentum considerations will be ignored.) With such replacements, k(T,P) can be calculated through eq 2-6. For the H CO case, the S matrix is indexed by all the energetically open vibrational and rotational states of CO. Each S element is a complex number containing an amplitude and a phase that describes the inelastic (or elastic) transition from the indexed initial to the indexed final state of CO induced by a single collision with H. Of particular relevance to the question of tunneling is the replacement of the RRKM expression for p ( E ) by one based on S. It can be s h o ~ n ~that l - ~p ~( E ) is given by
+
(38) Miller, W. H. J . Am. Chem. SOC.1979, 101, 6810. (39) For recent reviews, see: (a) Bowman, J. M. Ado. Chem. Phys. 1985, 61, 115. (b) Bowman, J. M.; Wagner, A. F.In Theory ofChemical Reacfion Dynnmics; Clary, D. C., Ed.; Reidel: Dordrecht, 1986; Chapter 3. (40) Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J . Phys. Chem. 1983, 87, 2264. (41) Bowman, J. M. J . Phys. Chem. 1986, 90, 3492. (42) Gazdy, B.; Bowman, J. M. Phys. Reo. Lett. 1987, 59, 3. (43) Kinsey, J. L. Chem. Phys. Left. 1971, 8, 349.
5320
The Journal of Physical Chemistry, Vol. 91, No. 20, 1987
Wagner and Bowman
where Q ( E ) is the Smith collision lifetime matrix44 given by Q ( E ) = i h S dSt/dE
HCO+H + C O
1
(13)
The invariance with energy of the unitarity of S can be used with eq 12 and 13 to produce an expression for p ( E ) that depends on the energy derivative of only the phase q5jj of each S, element:
To proceed further, qualitative features of the S near an isolated molecular resonance must be described. One can show that each S element will exhibit rapid characteristic changes as a function of total energy only immediately in the region of a molecular resonance energy and only to the extent that the inelastic transition represented by the element involved scattering that sampled the resonant molecular state in the well of the potential energy surface. Thus, S can be decomposed into a resonance, S', and a nonresonances portion, S"'
s = S' + S"'
T.Phys. Reo. 1960, 118, 349.
1.1
1.2
1.3
1.5
,'.I 4
1.6
H + C O + HCO
e 8
1.4
1.7
1
4
(15)
where all the energy-dependent changes in S in the vicinity of the resonance are isolated in S'. The amplitude of S' measures the extent of the resonance contribution to the i --j transition, and its phase will change by 2a as the energy is varied through the resonance. The phase for the corresponding element for S is a weighted sum of the combination of the phases of S' and S"'where the weights are approximately proportional to the ratio of the amplitude of the S' or S"' element to the amplitude of the S element. Through eq 14, it is the change in phase of the S elements that affects p ( E ) . The involvement of tunneling processes in an isolated resonance is reflected in the relative amplitude of S'. This in turn affects how much the change in the phase of S with energy is equal to the change in phase of S' with energy. Through eq 14, this affects p(E). In the limit of no tunneling, S' can dominate S"' and it can then be ~ h o w n ~ that * * ~p*( E ) becomes a delta function under several limiting forms of the energy dependence of the phase. The delta function form is what is used in RRKM theory. However, in the limit of significant tunneling, the total phase of S does not change the full 2 a as in a pure resonance condition but some fraction of 2 a depending on the relative size of the amplitude of S'to that of S"'. Then, under the same limiting conditions, p ( E ) becomes a weighted delta function where the weight is proportional to the amplitude of S'. In the replacement eq 12, p ( E ) becomes formally a dynamic quantity. It is, as in RRKM theory, centered on resonance energies. However, where the p ( E ) in RRKMn theory is a spectroscopic quantity with an integrated weight for each resonance state strictly equal to the state degeneracy, the p ( E ) of eq 12 scales in integrated weight with the dynamic accessibility of the resonance state (Le., the relative size of SI). Where the p ( E ) of RRKM theory is blind to whether tunneling is required for the participation of the resonance state in the dynnamics, the p(E) of eq 12 is reduced in size by the necessity of tunneling. If p ( E ) has an integrated weight that is dynamics based, then tunneling can be made consistent with the low-pressure limit. When such a p ( E ) is applied to the low-pressure limit of eq 10, it is still true that the integrand contains a strongly pressure dependent factor in the tunneling region of energy because of the ratio of a k,[M] of classically allowed size to a k ( E ) of classically forbidden size. However, that factor is now weighted in the integrand by a p ( E ) that is also of classically forbidden size. On the other hand, when the energy rises above the tunneling region, both p ( E ) and k(E) become classically allowed and can dominate pressure-dependent k,[M] in the integrand. For a traditional RRKM theory to qualitatively imitate this new theory, tunneling processes would have to be included in a way that modifies both p ( E ) and CP(E,). Otherwise, in most cases involving isolated (44) Smith, F.
1
0.5
1.0
1.5
2.0
2.5
3.0
3.5
lOOO/T Figure 4. Calculated low-pressure addition rate constant as a function of temperature for He buffer gas. Calculations with and without the thermally averaged J approximation are shown.
resonances, it would be best for traditional RRKM calculations to ignore tunneling altogether. The fact that experiment also demands such an application of traditional RRKM theory supports the qualitative form of the new theory. The calculations in progress using this new theory will shed further light on experimental verification of its form. B. Angular Momentum Summation. In typical applications of RRKM theory, the treatment of total angular momentum is handled by using a thermally averaged angular momentum, ( J ) , as discussed earlier. Because of the sparse set of states in HCO*, this approximation is not accurate at all temperatures. As the temperature increases, the value of ( J ) also increases. This in turn results in a greater increase in centrifugal energy at the bottom of the H C O well (where the external rotational spacings are large) than at the top of the addition barrier (where spacings are small). Consequently, the number of vibrational states that can be supported within the H C O well below the barrier to addition (or dissociation) decreases with increasing ( J ) or, equivalently, increasing T. Thus vibrational states, which at lower values of T (and ( J ) ) are bound, will at some specific value of T (and ( J ) ) become unbound, Le., above the barrier to addition in the effective potential. Since in H C O the density of unbound states is so low and since only unbound states are used in the RRKM calculations, at the temperature at which a formerly bound state suddenly becomes unbound, the computed rate constant can undergo a noticeable discontinuity. This is illustrated in Figure 4 for the calculated addition rate constant at a fixed pressure of 300 Torr and the calculated dissociation rate constant at a fixed pressure of 2.5 Torr in He buffer gas. For each rate constant, the standard calculation with explicit summation over J is compared to a calculation different only in the use of the ( J ) approximation. In general, this approximation is quite good. However, at a temperature of about 750 K, there is a jump in the rate constants which is caused by a new molecular vibration state becoming unbound in the effective potential for the temperature-dependent ( J ) . In the full summation over J , all values of J are presented at every temperature but are weighted by the Boltzmann factor. Thus, all molecular vibration states are present at all temperatures but become important only when their Boltzmann factor is significant. As discussed earlier, the standard calculation in Figure 4 has a ( K ) approximation to replace the rigorous summation in the K projection quantum number. Because of this approximation, just as in the case of the ( J ) approximation, not all of the unbound molecular states are present at all temperatures. At different
H
+ CO
S=
The Journal of Physical Chemistry, Vol. 91, No. 20, 1987
H C O Reaction
specific temperatures for each value of J, unbound molecular states suddenly appear in the calculation due to the increasing value of ( K ) . This is a much smaller effect than that due to ( J ) alone, and it is spread out over all the J terms at different temperatures. The result is that the calculations have no large scale discontinuities, as in Figure 4 due to the ( J ) approximation. However, some fine scale ripples in the calculation still occur due to this effect, as can be seen in the addition results of Figure 4 (around 1000/T = 1.5) and in several subsequent figures. The difference in the two calculations in Figure 4 and the effects just mentioned for the K projection quantum number are strongly accentuated by the exclusion of tunneling states discussed previously. In the thermal average approximation, some of the bound molecular states at lower values of T (and J and K ) are unbound if tunneling is considered. As T (and J and K ) increase, these molecular states have energies closer to the top of the effective barrier to addition (or dissociation) and thus require less tunneling to become unbound. If, as in the S matrix theory discussed above, the contribution of tunneling states in the low-pressure thermal dynamics were weighted by the degree of tunneling, then the discontinuous changes in the RRKM rate constant at specific temperatures would be distributed over a range of temperatures according to the diminished tunneling required as the molecular state energy approaches the top of the addition barrier. Thus unlike RRKM theory, a thermally averaged J approximation in the S matrix theory might be reasonably accurate at all temperatures. C. Energy Parameter Sensitivity. For direct reactions that involve no metastable complex, there is usually a Boltzmann exponential dependence of the rate constant on the barrier to reaction. This is the basis for the Arrhenius expression for such rate constants. In the low-pressure limit for addition or dissociation reactions, it is well-known3' that this Boltzmann exponential dependence is reduced as can be seen by rewriting eq 10 in the exact low-pressure limit
where E , is the zero-point corrected asymptote for dissociation kcal/mol). As long as E , is fixed, the (Le., - A H ( O K)H+CO-HCO, entire dependence of the v is in the lower limit of the integral. Since the integrand in eq 16 is always positive, if p ( E ) is a continuous function of E, increasing V'! will always decrease the rate constant. If p(E) is a constant, it is easy to show that the decrease in the rate constant goes as exp(-V/kBT), the Arrhenius factor typical of direct reactions. If p ( E ) is a continuously increasing function of E, as is generally the case, the rate constant dependence on fl will be reduced by a factor approximately proportional to p ( E , + p)in most cases (Le., the increasing density of states partially compensates for the decreasing Boltzmann factor at the top of the barrier). In the isolated resonance model, the dependence of p can become discontinuous. For isolated resonances, the integral in eq 16 is converted into a summation over the resonance energies. Then any change in fl that causes E , + v to fall between the lowest resonance state included in eq 16 and the highest molecular state not included in eq 16 will not change the number of resonance states or their weight included in the summation. Thus, small variations in will have no effect on the low-pressure limit. Larger variations in p will cause discontinuous changes in the rate constant as new resonance states are added (fldecreased) or old resonance states eliminated (pincreased) from the summation. If E , is varied while v is fixed, a continuous representation of p ( E ) will result in no change in the rate constant if p ( E ) is constant and a change approximately proportional to p ( E , fl) if the density of states is increasing. However, in the isolated resonance model, small variations in E , will produce an exact exponential variation in the rate constant because, according to eq 16, every resonance state included in the summation will have its Boltzmann weight changed. Larger variations that result in terms being added or subtracted to the summation will produce
+
0.0
5321
I
1
/
7.0
5.0 -
6
g.3
4.0-
DISSOCIATION
Y
3.0
-
20 1.0
ADDITION 5
Figure 5. Ratio of the calculated low-pressure dissociation and addition rate constants for HCO and DCO vs. temperature. Calculations with the isolated resonance form and with the Whitten-Rabinovitch form of the density of states are represented.
discontinuous changes in the rate constant. Equation 16 approximates the effect of total angular momentum summation. With rigorous inclusion of this summation, perfect independence of the rate constant for f i or perfect exponential dependence of the rate constant on E , requires that small varip fall between state energies for every effective ations in E , potential dependent on J. In practice, over the temperature range of room temperature to about 700 K (temperatures below the break in Figure 4),the ideal dependence of k( T,P) on p and E , is maintained to within a few percent for variations of a few tenths of a kcal/mol in either quantity. D. Isotope Effects. Hydrogenic isotope effects measure the relative difference of H and D participation in the reaction dynamics. If the isolated resonance model predicts H/D relationships different from that of either overlapping resonance models or tunneling models, then isotope effects become a measurable way to test the isolated resonance model. Some such measurements are available for dissociation and are presented for the first time in paper 2. Calculated results will now be presented that both qualitatively agree with the measured dissociation isotope effects (see paper 2) and suggest measurements for addition isotope effects (none are available) which would be able to test the isolated resonance model. In Figure 5 the isotope effects on the addition and dissociation rate constant as a function of temperature are presented both for the isolated resonance model calculation and for a calculation that uses the Whitten-Rabinovitch continuous representation of p ( E ) . In these calculations, the pressure for addition is set at 300 Torr and that for dissociation at 2.5 torr. (The pressures are typical of experimental measurements discussed in paper 2.) However, the conditions are sufficiently close to the low-pressure limit that there is negligible pressure dependence of the calculated isotope effect. This implies that the isotope effects are only sensitive to the density of states of the molecule and not to the sum of states at the transition state. Consider first addition. The low-pressure form of the rate constant, eq 16, applies, and the isotope effect can be conveniently divided up into three parts: effects from the partition functions before the density integral (Le., changes in reduced mass and moments of inertia), effects on f i (due to zero-point energy changes) in the lower limit of the integral, and changes in the density integrand itself. From Table I, the partition function effect for addition can be shown to favor H over D addition by a factor of 1.68. If p ( E ) were a constant with energy and the same for either H or D reactions, then the p effect combined with the
+
5322 The Journal of Physical Chemistry, Vol. 91, No. 20, 1987 partition function effect still favors H over D addition by a factor of 1.4 at room temperature. The results in Figure 5 show that the isolated resonance model favors H over D addition at room temperature while the Whitten-Rabinovitch calculation does not. In the latter calculation, the greater density of states for DCO and its steeper energy dependence have overcome the basis that the combined partition function and the p effects impose. However, in the isolated resonance model, even though there are more DCO than H C O resonance states, the effect is not enough to overcome the initial bias. There are several reasons for this. First, the fl effect disappears in the isolated resonance model for small isotopic variations (in H (D) + CO, the variation is 0.1 kcal/mol) for the reasons discussed in the previous part of this section: any variation of within the spacings of the resonance levels of HCO and DCO is not detected by the calculation. Thus, the stronger H bias imposed by just the partition function effect must be overcome by the density. Second, each state for both molecules comes in with the same delta function weight. Therefore, what matters is the resonance energies over which, in the isolated resonance form of eq 16, one sums a Boltzmann weight as measured from the addition asymptote. The results in the figure indicate that there are enough H C O resonance energies close enough to the lowest DCO resonance energies to keep the H summations smaller but still comparable to the D summations. For dissociation, both kinds of calculations are dominated by a exponential behavior with temperature that reflects the larger dissociation energy for DCO than H C O because of zero-point energy effects. However, at any temperature, the dissociation and addition isotope effects are related in both calculations by the same value for the ratio of equilibrium constants. Hence, as in addition, the dissociation isotope effects are smaller in the Whitten-Rabinovitch model than in the isolated resonance model. This basic result in Figure 5 that the isolated resonance model predicts larger isotope effects than the Whitten-Rabinovitch model does not change with variations in either fl or E,. Such variations do not significantly change the specific value of the isotope effect in the Whitten-Rabinovitch model but can substantially change the effect in the isolated resonance model. In particular, the isolated resonance calculations in paper 2 that best agree with all available measurements predict an isotope effect of 0.95, a value below unity unlike the result in Figure 5 . The same calculation with the Whitten-Rabinovitch model gives a value essentially identical with that in Figure 5 of 0.81. At lower temperatures, the differences between the isotope effects for these two kinds of calculations increases. While it is experimentally quite difficult to study dissociation at room temperature, addition rate constants have been measured only at or near room temperature. Hence, the results in Figure 5 would suggest that precise measurements at low pressure and at room temperature that could distinguish isotope effects of 0.8 from those of unity or higher would distinguish the isolated resonance and Whitten-Rabinovitch models. This level of precision is difficult to obtain experimentally at present. In the first part of this section, arguments are presented that show that tunneling is inconsistent with pressure dependence effects. A more traditional measure of tunneling is the temperature dependence of isotope effects. In the low-pressure limit, the inclusion of tunneling implies the inclusion of resonance states behind the addition barrier (dotted lines in Figure 1). Away from the low-pressure limit (see E!q lo), the inclusion of tunneling affects both what resonance states are considered and the k ( E ) for those states. The resulting pressure-dependent isotope effects can be complicated. In Figure 6, the effect of tunneling on the calculated isotope effects in the isolated resonance model are displayed. As in Figure 5 , for the calculations without tunneling, the pressure for addition is set at 300 Torr and that for dissociation at 2.5 Torr. However, for the calculations with tunneling, three pressures are presented, including an extremely low pressure which guarantees that the low-pressure limit has been reached even for the tunneling case. At the low-pressure limit, the inclusion or absence of tunneling
Wagner and Bowman
7.06.0-
NO TUNNWNG
____.______._.._.___~..---------. TUNNELING P = 300 torr ........ N.... N............. N W N........ G P...=.................. 2.5 torr ........... N N N-WN G P =- __ 10-a torr- -
5.0 -
4.0 Y
3.0 2.0 -
1.0
ADDmON
.5
Figure 6. Ratio of the calculated dissociation and addition rates for HCO and DCO vs. temperature. Isolated resonance calculations with and without tunneling are represented. The calculations without tunneling are essentially at the low-pressure limit (see text). The calculations with tunneling are at the listed pressures.
does not change the qualitative trend that H reaction is enhanced over D reaction. However, the enhancement is far greater with tunneling than without. As the pressure is increased, the enhancement is reduced, first at the lower temperatures and then at the higher temperatures. By 300 Torr, at room temperature D addition is essentially the same as H addition. The results of Figure 6 suggest that the strong pressure sensitivity of isotope effects makes pressure dependence of the rate constant for either isotope a better marker for tunneling than the temperature dependence of the isotope effect. E . Recrossing. In the calculations presented so far, the calculated rate constants have been at or near the low-pressure limit. As paper 2 will show, all the rate constant measurements are in this region. The low-pressure limit, via eq 10 or 16, emphasizes the role of the density p ( E ) in controlling the dynamics. On the other hand, the high-pressure limit emphasizes the role of the sum of states C P ( E , ) in the dynamics. As discussed in the previous section, an R k K M theory based on isolated resonances allows only resonance width slices of C P ( E , ) as a function of E to contribute. The unselected portion of CP(E,) must be attributed to recrossing. In what follows, the approach to the high-pressure limit will be investigated as a measure of recrossing in the HCO reaction system. From eq 5, in the limit of high pressure, the rate constant for the isolated resonance model takes the form
From eq 3, the XP(E,) is contained in the formula through f ( E , ) . In the usual high-pressure limit for a continuous function of p ( E ) (Le., that derived from eq 2), the sum over resonance width slices of f ( E ) in eq 17 is replaced by the much larger integral over all off(E). The difference between the sum and the integral is the measure of recrossing. Unlike the case in the low-pressure limit, the actual value of the rate constant does depend on the specific resonance widths LIE#. The results in Figure 7 illustrate the changes in the highpressure limit produced by an isolated resonance model. In the figure, three calculated rate constants for addition at room temperature are displayed over a large range of pressure that includes both the low- and high-pressure limits. An insert in the figure shows the same three calculated rate constants in the low-pressure limit as a function of temperature. One of the calculations in the
H
+ CO
The Journal of Physical Chemistry, Vol. 91, No. 20, 1987 5323
H C O Reaction H
+ CO
HCO
Resonance
0.5
1.5
2.5
3.
1ooo/r 0.0
1.0
20
3.0
4.0
5.0
6.0
7.0
M P ) (torr) Figure 7. Calculated addition rate constant as a function of pressure at room temperature for He buffer gas. The insert displays calculated low-pressure addition rate constants as a function of temperature. Calculations with a delta function representation and a Whitten-Rabinovitch representation of p ( E ) are shown.
figure is the isolated resonance model appropriate for HCO with AEi set to 1 cm-'. The other two are calculations based on the Whitten-Rabinovitch approximation to p ( E ) . They differ in the values for fl and ( A€)tot. One of these two calculations has the same values as in the isolated resonance calculation. For the other, the values have been adjusted so that the low-pressure limit over an extended temperature range is identical with the one for the isolated resonance calculation. To do this, v had to be lowered by 0.3 kcal/mol and (AE)tolhad to be raised by 20 cm-'. Consider first the larger plot of pressure dependence. The Whitten-Rabinovitch calculations, whatever their inaccuracies in estimating p ( E ) , employ a continuous representation of p ( € ) andflE) [or C P ( E , ) ] as functions of energy. Hence, they have no recrossing built in and assume the high-pressure limit expected of a conventional RRKM calculation. Figure 7 clearly shows that the recrossing feature of the isolated resonance calculation has lowered the high-pressure limit by 2 orders of magnitude from that for either calculation based on the fullf(E). In the isolated resonance calculation, the high-pressure limit is reached by about 100 atm, but for the other calculations about 2 orders of magnitude higher pressure is needed to reach the limit. The great difference between the continuous and isolated resonance methods of calculating the Boltzmann-weighted sum of states at this temperature is due to the very sparse spectrum of resonance states of H C O (as estimated by the harmonic oscillator model). Presumably, as the temperature increases this difference would be attenuated as the denser part of the resonance spectrum would make a greater contribution to the Boltzmann-weighted sum of states. The insert in Figure 7 shows that a calculation based on a continuous representation of p ( E ) can produce low-pressure rate constants essentially identical with those of an isolated resonance model provided slight adjustments in and (A€),,, are allowed. The agreement persists from room temperature to over 1000 K and could probably be extended if the temperature dependence of ( A E ) , o ,were also adjusted. Without adjustments, the differences in the low-pressure limits between the Whitten-Rabinovitch and isolated resonance calculations are substantial enough (35% at room temperature) to be experimentally distinguishable. However, there is no independent way to know the values of v or (AE)lotwithin the margin of error of these small adjustments. Thus, realistically, uncertainties in P, the value and temperature dependence of (AE),,,, and experimental uncertainties in any rate constant measurements make it unlikely that the full p ( E ) and the isolated resonance model can be confidently distinguished by
low-pressure studies alone. (As mentioned earlier, low-pressure isotope effects, not the absolute value of any one rate constant, offer the best hope of distinguishing theoretical models in the low-pressure limit.) On the other hand, as the larger plot in Figure 7 shows, high-pressure studies at one temperature clearly distinguish calculations that are indistinguishable in the low-pressure regime. As paper 2 will show in more detail, the isolated resonance calculation in the figure is very similar to one which is fully compatible with all the low-pressure measurements (both addition and dissociation) available for the buffer gas He. In this sense, then, the results in Figure 7 constitute a prediction of what the highpressure limit of HCO is and where it will occur. Unfortunately, the high-pressure limit depends directly on the resonance widths A€, while the low-pressure limit does not. If the resonance widths were increased to 10 cm-' or decreased to 0.1 cm-I, the highpressure limit of the isolated resonance model would go up or down accordingly. Whatever the case, the isolated resonance nature of HCO will guarantee substantial reduction in the high-pressure limit. Thus, the difference between the fullf(E) calculations, even with the adjustments of v and ( AE),,,, and the isolated resonance high-pressure limit suggests that an experiment at one temperature should be able to qualitatively distinguish these two calculations and directly measure the role of recrossing and isolated resonances in the dynamics. F. Stabilization with Isolated Resonance States. Most of the rate constant calculations shown and all that can be directly compared to experiment in paper 2 are at or near the low-pressure limit. The effect of the buffer gas via k, is an important part of these calculations. The method used for constructing k, is due to Troe3' and is based on an approximate analytic solution to the master equation for the dissociation of a molecule via repeated collisions with a buffer gas. A comparison of the master equation solution to the RRKM solution identifies what k, should be in order to mimic the master equation solution. The result is an analytic expression for a stabilization efficiency p, which, when multiplied by the gas kinetic rate of buffer gas-metastable molecule collisions, gives k,. The analytic master equation solution used in this procedure is derived under the conditions that (1) the thermally averaged probability of energy transfer in a molecule-buffer gas collision is governed by an exponential gap model and (2) p ( € ) has the form e-pE near the dissociation energy. This latter condition is violated by the isolated resonance model for HCO. The effect of removing this assumption will now be examined as it applies to HCO. The form of p ( E ) and the exponential gap form of the probability are coupled in the derivation. The exponential gap model ensures that the probabilities are governed by two types of exponential parameters, say a and 6, one controlling energy loss from the molecule and one controlling energy gain. These two parameters should be a function of the initial energy of the molecule, but this functionality and the two parameters are all linked by time reversibility: p ( E ) times the probability of energy gain to E'via a collision must equal p(E? times the probability of energy loss to E . If p(E) is exponential in the energy, Troe showed that the initial energy dependence of a and 6 disappears and a and 6 at all initial energies obey a very simple relationship that depends only on the p term in p ( E ) . This permits an analytic solution to the master equation and ultimately produces a (3, expression that depends only on a and 6. Since the total average energy transfer (A€),,,, a measurably quantity, also can be expressed in a and 6 from the probability of energy transfer, p, can be expressed in final form in terms of measured values of ( A E ) , , , and, implicitly, RRKM calculated values of p. In an isolated resonance model, p ( E ) is not an increasing continuous function of energy but is discrete. The best match of the analytic model of Troe to this case would be to eliminate the energy dependence of p ( E ) by setting p equal to zero. However, the spacings between isolated resonance states in HCO, generally larger than typical values of (A€),,,, can perhaps significantly alter the correct value of (3,. To examine this issue, a numerical
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The Journal of Physical Chemistry, Vol. 91, No. 20, 1987
0.35
-
300 K
0.25
Q2 0.20 0.15
I
0.00 ~
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
1
0
-+,,, (cm-’) Figure 8. Gas kinetic stabilization efficiency p, vs. ( A E ) , , , at 300 K. Both the continuous analytic solution and the isolated resonance numerical solution are shown. See text for details.
solution to the master equation was obtained by using the HCO isolated resonance density of molecular states in the time reversal constraint but otherwise an exponential gap law for the probability of transitions between the molecular resonance states. The numerical calculations are the isolated resonance analogue for HCO of the analytic model. The results of the numerical calculation and the analytic model (for p = 0) are displayed in Figure 8 in terms of the value of p, vs. the input value of ( The figure indicates that the isolated resonance model causes a lowering of (3, for a fixed value of but only by about 10% or less. The results i n Figure 8 do depend on the energy spacings between the resonance states. If the energy spacings are on the order of a factor of 2 larger than that used in the figure, the isolated resonance values of p, rise to near the maximum value of 1 .O. However, for smaller energy spacings (such as those in DCO), the isolated resonance values of p, are similar to those shown in the figure. As Figure 8 shows, for H C O the isolated resonance lowering of stabilization efficiency is a small effect. Since there are many approximations in the master equation solution (including those not discussed regarding the interconversion of rotational and vibrational e n e r g ~ ~upon ~ , ~ collision ’ with the buffer gas), the improvement i n accuracy from a numerical master equation solution is not significant. Consequently, the more convenient analytic solution was used in all other calculations in this paper and paper 2. V. Summary In this paper, RRKM calculations have been carried out on the thermal addition of H C O and the thermal dissociation of HCO in He buffer gas. The calculations have been carried out on the Harding surface for HCO which is based on ab initio electronic structure calculations whose energetics at critical points
+
have been empirically modified. The extremely sparse set of molecular vibration states for H C O has motivated an extreme form of RRKM theory in which all the reaction dynamics takes place through isolated molecular vibrational resonances. Rate constant calculations with the model show the following characteristics: 1. Tunneling cannot be included in the calculations without unrealistic values for the collisional stabilization efficiency of HCO* by buffer gases. By comparison to a recently developed scattering matrix theory of addition and dissociation, this exclusion is rationalized as a consequence of a fundamental flaw in the RRKM spectroscopic, rather than dynamic, definition of the density of molecular states at energies where tunneling dominates. 2. Explicit summation over total angular momentum is required to avoid discontinuities in the rate constant as a function of temperature. Similar discontinuous changes can occur as a function of changes in the addition barrier or dissociation barrier. 3. Isotope effects in the low-pressure limit favor H over D dissociation or addition more so in the isolated resonance RRKM model than in more traditional (but for HCO, theoretically arbitrary) RRKM models. This is due to the fact that the reduced density of states in the isolated resonance models also affects the relative density of states between HCO and DCO. Precise isotope effect measurements for addition at room temperature would be a test the isolated resonance model. 4. Rate constant measurements approaching the high-pressure limit should be able to distinguish between the isolated resonance RRKM model and more traditional (but for HCO, theoretically arbitrary) RRKM models. This is due to the high degree of recrossing predicted for the isolated resonance model alone. Low-pressure rate constant measurements (except for isotope effects noted above) probably can not unambiguously distinguish these models because of uncertainties in a number of quantities, including the average energy loss per collision and the barrier to addition. In this paper, there has been no comparison to previous measurements, all of which are in the low-pressure limit. In the following paper, these measurements are reviewed, new measurements on dissociation are presented, and a full comparison of theory to experiment is offered. These comparisons are generally quite favorable and indicate the isolated resonance model discussed here is consistent with all the available experimental data. As has been indicated throughout this paper, the RRKM calculations presented here are based on simplistic descriptions of molecular resonances that only detailed dynamics studies can remedy. Such calculations are in progress. The success of this RRKM study, especially as detailed by comparison to experiment in the next paper, suggests that these more complicated calculations will be consistent with experiment.
Acknowledgment. The authors especially thank L. Harding for helpful discussions and auxiliary calculations of characteristics on the potential energy surface. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, U S . Department of Energy, under Contract W-3 1- 109-Eng-38. Registry No. H, 12385-13-6; CO, 630-08-0; HCO, 2597-44-6; D,, 7782-39-0.
