J . Phys. Chem. 1985,89, 5289-5294
5289
Rotat lonal Excltatlon in Low-Energy CH,CN-He Collisions Sheldon Green N A S A Goddard Space Flight Center, Institute for Space Studies, New York, New York 10025 (Received: July 1 , 1985) Extensive coupled states (CS) scattering calculations were done for low-energy CH$N-He collisions, using a theoretical interaction potential, to obtain state-to-state rotational excitation rates at 40 K. These were used to test the accuracy of the infinite-order sudden (10s) scattering approximation,the 10s scaling law, and also the "energy-corrected sudden" (ECS) corrections to the latter. The 10s scattering approximation gives a reasonable estimate for the fundamental rates, R(0,O -J,k), although, as for linear rotors, for highly inelastic collisions a correct interpretation of the 10s energy is important. With CS values for RG,k 0,O) as input, 10s scaling gives reasonable estimates for the entire matrix of state-to-state rates, especially for those transitions in which k does not change, where the accuracy is typically f20%; although the accuracy for transitions across k ladders is somewhat less good, it has not been possible to pinpoint the source of this problem due, in large part, to a lack of adequately converged CS rates to some of the higher rotational levels. The ECS corrections were not found to be useful.
-
Introduction Although the exchange of rotational and translational energy in molecular collisions has been extensively studied, both experimentally and theoretically, for the most part this work has concentrated on the simplest case, collision of a linear molecule with an atom. That case now seems to be reasonably well understood. For systems where the intermolecular forces have been obtained from a b initio calculations and collision dynamics have been treated within the converged close coupling formalism, excellent agreement has been obtained with a wide variety of experimentally observed phenomena. Reasonable methods for making (semiempirical) estimates of intermolecular forces have been developed; and scattering approximations, notably the coupled states' (CS) and infinite-order sudden2+ (10s) approximations, have been shown to be accurate and computationally tractable. The 10s approximation precicts simple scaling relationships among different cross sections.5 The simple form of these scaling relations remains when first-order corrections are made, giving the energy-corrected sudden6 (ECS) method. A wide variety of experimental rates have been shown to follow the 10s or ECS scaling.' All of these techniques can be extended, in principle, to more complex systems, but only a few cases have been studied to date. The present paper examines collisions of helium atoms with a rigid symmetric top rotor, methyl cyanide. This work is motivated partly by the need for collisional excitation rates in this system to analyze radioastronomical data.* The only other symmetric top collision system studied to date is NH3-He which is complicated by the vibrational inversion of ammonia which lifts the *k degeneracy of the symmetric top rotor A nearsymmetric top system that has been studied is H2CO-He.'2 Interaction Potential As discussed in ref 9, the interaction potential depends on the position of H e relative to the center of mass of CH3CN and is most conveniently expressed in polar coordinates V(R,B,+). For consistency with the rotational basis functions used in the scattering calculation, B = 0 is chosen along the symmetry axis (along (1) McGuire, P.; Kouri, D. J. J. Chem. Phys. 1974, 60, 2488. (2) Pack, R. T J. Chem. Phys. 1974, 60, 633. (3) Secrest, D. J. Chem. Phys. 1975, 62, 710. (4) Hunter, L. W. J. Chem. Phys. 1975, 62, 2855. (5) Goldflam, R.; Green, S.;Kouri, D. J. J. Chem. Phys. 1977, 67, 4149. Goldflam, R.; Kouri, D. J.; Green, S. J. Chem. Phys. 1977, 67, 5661. (6) DePristo, A. E.; Augustin, S. D.; Ramaswamy, R.; Rabitz, H. J. Chem. Phys. 1979, 71, 850. (7) Brunner, T. A,; Pritchard, D. Adu. Chem. Phys. 1982, SO, 589. (8) Cummins, S. E.; Green, S.; Thaddeus, P.; Linke, R. A. Astrophys. J . 1983, 266, 331. (9) Green, S. J. Chem. Phys. 1976, 64, 3463. (10) Green, S. J. Chem. Phys. 1979, 70, 816. ( 1 1) Green, S. J. Chem. Phys. 1980, 73, 2740. (12) Green, S.; Garrison, B. J.; Lester, W. A.; Miller, W. H. Astrophys. J . Suppl. Ser. 1978, 37, 321.
C-C-N) and the xz plane (6 = 0) is chosen as a reflection plane passing through one of the hydrogen atoms. Since values of the potential are needed at a large number of distances and orientations, accurate ab initio calculations become quite expensive. A primary goal of the present study is to determine the accuracy of approximate scattering methods and this does not depend critically on the verisimilitude of the potential. Therefore, the relatively inexpensive G~rdon-Kim'~-'~ electron gas approximation was used. Although it has known shortcomings, it gives a reasonable qualitative picture of the short-range forces likely to be important for rotational excitation in this system so that the resulting rate constants will be a t least qualitatively correct. The potential was calculated at a set of 45 angular orientations for intermolecular distances of R = 3.5(0.5)8.5ao (1 a. = 1 bohr radius = 5.29 X lo-" m). For the scattering calculations the orientation dependence is expanded in spherical harmonics
and continuous radical functions are obtained by interpolating the vb(Ri).I5 To eliminate spurious behavior in the fitted potential it was necessary to generate points at additional distances. At each orientation the potential is a smooth function of distance, and it was convenient to get values at distances midway between calculated values by spline interpolation; also, the potential was extrapolated into the classically forbidden region (to R = 2 . 5 ~ ~ ) by assuming exponential repulsion at short range. The resulting points were fit to the 27 terms in eq 1 with h 6 11, p = 0, 3, 6 by minimizing the sum of the squares of the relative deviations. This fit gives average relative deviations that vary from 20% on the repulsive wall to a few percent in the region of the minimum. The most serious failure of the fit is its inability to reproduce the very steep repulsion predicted for close approach of the He to the methyl hydrogen atoms. Finally, values of uAr (R= 8.5) were adjusted so that extrapolation to long range, based on the last two radial points, would have reasonable inverse power behavior. Numerical values of the radial coefficients used in the scattering calculations are tabulated in supplementary material accompanying this paper.
CS Scattering Calculations The CS scattering formalism is presented in detail in ref 9. The two parity levels e = 1 and e = -1 are degenerate, and only the standard degeneracy averaged (summed over final and averaged over initial) values are presented here. It is readily shown that identical results are obtained by using a "primitive" symmetric ( 1 3) Gordon, R. G.; Kim, Y. S. J. Chem. Phys. 1972, 56, 3 122 (14) Green, S.; Gordon, R. G. POTLSURF,Quantum Chemistry Program Exchange, University of Indiana, Bloomington, IN, Program 25 1. ( 1 5) Green, S. J. Chem. Phys. 1977, 67, 71 5
This article not subject to US.Copyright. Published 1985 by the American Chemical Society
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The Journal of Physical Chemistry, Vol. 89, No. 24, 1985
TABLE I: Dependence of auk j k j'k' 00 10 20 30 70 80 12 0 13 0 3 3 53 63 10 3 3 43 53 63 3 3 43 5 3 63
-
Green
j'kq on CS Basis Set"
basis set
A-9-5
A-13-6
A-15-7
A-15-7-6
A- 17-9-6
A-19-1 1
5.3 1 21.60 11.53 6.70 1.43
5.97 20.54 11.99 5.12 1.85 0.19 0.39 1.42 1.20 0.57 1.37 1.20 1.24 0.6 1 23.28 9.66 4.20
5.83 20.79 11.39 4.90 2.05 0.39 0.50 1.26 0.99 0.55 1.47 1.59 0.93 0.57 21.96 5.88 3.98
5.83 20.84 11.36 4.90 2.05 0.40 0.50 1.25 0.98 0.55 1.47 1.59 0.93 0.57 22.10 5.86 4.02
6.21 21.23 11.17 5.27 2.72 0.44 0.23 1.28 0.58 0.65 1.31 1.54 0.75 0.57 21.50 4.43 1.88
6.19 21.07 11.79 4.73 2.58 0.34 0.24 1.16 1.18 0.32 1.29 1.44 1.03 0.54 21.55 4.76 2.67
0.65 1.15 1.12 1.49 1.29 16.52 10.18
"Calculations were done at a total energy of 60 cm-' and included partial waves through I = 20. Values are in top basis, in which k takes both positive and negative values, and then summing and averaging over degenerate k levels. The energy levels for the symmetric top rotor are given by E ( j , k ) = (1/2)(A
+ B ) u ( j + 1) - k2] + Ck2
(2)
where the rotational constants for methyl cyanide are A = B = 0.3068 cm-' and C = 5.2470 cm-I. Because of the nuclear spins of the equivalent hydrogen atoms the levels separate into two noninteracting symmetry sets, just as in ammonia: A levels which have k = 3n and E levels which have k = 3n f 1. Although the rotational energy increases quite rapidly with k , the spacings between successivej values within a k ladder are small, and despite the separation of A and E symmetries, the number of levels accessible even at low temperatures prohibits the use of close coupling calculations. Fortunately, the CS approximation is generally found to be accurate for systems dominated by shortrange forces as is the case here. In particular, CS was found to be accurate for NH3-He collision^.^ A series of calculations was done to test convergence of CS cross sections on increasing the rotor basis set. It is convenient to label the basis sets for A levels as A-jo-j3-j6where j , is the highest j value for k = 0, j 3 is the highest j value for k = 3, etc; E level basis sets are labeled analogously E-jl-j2. For the A levels, calculations were done at a total energy of 60 cm-I. Convergence as a function of basis set size is shown in Table I for selected state-testate cross sections. The basis set A-1 3-6 includes all open channels. The largest basis set, A-19-11, requires 38 coupled channels. It is seen that k = 6 levels, all of which are closed at this energy, are not important. The cross sections are not so well converged, however, with respect to the closed k = 0 and k = 3 channels, and it appears that uncertainties of typically 20%-with larger errors in cross sections that involve the highest levelscannot be avoided with computationally feasible basis sets. Final calculations for cross sections used basis A-17- 11 at energies of 150, 100, and 80 cm-I; A- 15-7 at 60,55, and 50 cm-I; A-15-5at 45 cm-I; A-14-5 at 40 cm-I; A-13-5 at 30 and 20 cm-I; and A-1 1-4 at 12 cm-'. For economy, calculations were generally limited to projections of rotational momentum on the body-fixed axis, j , 6 4,so that cross sections are not valid where both J and j'are greater than this value. Rate constants at a kinetic temperature of 40 K were obtained by integrating over a Boltzmann distribution of collision energies, assuming the cross sections vary linearly with energy between computed points. Cross sections often show strong resonance behavior at low energies. This can require calculations a t many energy values in order to obtain an accurate Boltzmann average. Also, the C S approximation may not predict the resonance structure accurately. In the present calculations the cross sections all appear to be relatively smooth functions of energy; it will be assumed that any resonance structure that has escaped detection owing to the coarse
A*.
energy grid used here does not significantly affect resulting rate constants. Similar calculations were done for the E levels. Basis set tests at 60 cm-I suggest that the k = 4 and k = 5 levels, all of which are closed, are not important. The basis of all open channels at this energy, E- 12-10, requires 42 coupled channels, and larger basis sets were too expensive to use in the production runs. Limited tests with basis sets to E-14-1 1 (48 channels) suggest that cross sections to the higher levels may have errors of up to about a factor of 2. Final calculations for cross sections used basis E-12-10 at energies of 200, 100,60, and 50 cm-I; E-12-8 at 40 and 35 cm-I; E-10-7 at 30 and 25 cm-I; and E-8-3 at 20, 17.5, 15, 12.5, and 10 cm-I. Rate constants were computed for a kinetic temperature of 40 K. 10s Scattering Calculations The 10s scattering formalism for symmetric tops is discussed in ref 10. On taking the average over degenerate levels the final working equations for rate constants are given as follows for k = k' = 0
R(jO - + j ' o ) = (2j' for k = 0 , k'
+ l)x(j'
jo)'Q(l,O)
I
(3a)
>0 R ( j 0 4 j ' k ' )= 2(2j'
+ l)Z(i,
-I k , L)'Q(I,k') (3b)
1
for k > 0 , k' = 0 R(jk-j'0)=(2j' for k
> 0 , k' > 0
R ( j k j j ' k ' ) = (2j'
+ 1)Z 1
+ l ) x[ (Oj '
k - k )'Q(l,k)
(3c)
{(i, k-
i k ) Q ( l , k - )+
(Ck.
+:
{ k ) Q ( I j k + ) (3d) }
where k- = k - k' and k+ = k + k'. The Q(l,k) here are Boltzmann averages of the Q'(/,k,k)defined in ref 10. The Q(Z,k) can be related to the rates into or out of the lowest, j = k = 0 , level as follows. Qo',k) = (2 - 6k30)-'R(0,0 j , k )
-
+
+
(4a)
Q(j,k) = (2j l)R(j,k 0,O) (4b) Recall that the 10s approximation ignores all rotational energy spacings so that forward and reverse rates calculated within the 10s framework will fail by a Boltzmann factor to satisfy detailed balance relations. 10s calculations were done to obtain the Q(L,M) at the same energies used in the C S calculations for the A levels. Note that the same set of Q(L,M) predicts rates among both the A and E
The Journal of Physical Chemistry, Vol. 89, No. 24, 1985
Low-Energy CH,CN-He Collisions
r
1
1
I
I
5291
K'f K
15
5tdLLA
- 1.0
-0.5
0
0.5
1.0
J
Figure 1. Comparison of CS and 10s excitation rates, R(0,O j , k ) , in CH3CN-He collisions. CS values are shown as dots for k = 0 and as crosses for k = 3. 10s values, using different interpretationsfor the 10s energy, are shown as solid lines for k = 0, dashed lines for k = 3, and a dotted line for k = 6 . For k = 0 and k = 3 the upper curves assume the 10s energy is the initial energy for upward transitions, RIU;the lower curves assume it is the initial energy for downward transitions, RID; and the heavier, middle curves give the geometric average of these, RIA. For k = 6 only RIA is shown. +
levels. The integrals over orientation were done by Gauss quadratures using 32 points for 19 and 4 points for d. Calculations with a 40 X 6 quadrature grid (at a collision energy of 40 cm-I) suggest that the coarser grid is accurate to better than 10% for L d 20, M d 6 .
Results In the absence of more rigous values it is necessary to assume that CS rate constants correspond to the true values for this system (at least for the intermolecular potential used here). This is not unreasonable as CS has been shown to be quite accurate for other systems including NH3-He and H2CO-He which are not too dissimilar to CH3CN-He. Some uncertainties must be anticipated, of course, due to the limited CS basis sets and the small number of energies in the Boltzmann averages. The C S values will be used to test (1) the accuracy of the 10s approximation, and (2) the validity of 10s and ECS scaling relationships. In the 10s approximation the interpretation of the energy is ambiguous. Chapman and Greeni6 recently considered taking the 10s energy as either the initial kinetic energy or the final kinetic energy. Because rates for forward and reverse processes still do not satisfy detailed balance, these energy interpretations are applied to either upward or downward transitions, with the rate for the reverse process then fixed by the detailed balance requirement. For rates out of the lowest level in the CO-He and CO-Ar systems it was found that taking the 10s energy as the initial energy for upward transitons, R I U gave , an overestimate, while taking the 10s energy as the initial energy for downward transitions, R I D gave , an underestimate. The geometric average (16) Chapman, S.; Green, S. Chem. Phys. Lett. 1984, 112, 436. Green, S.; Chapman, S. Chem. Phys. Lett. 1983, 98, 467.
-
40
--
K'= K
24
-
20
4d
-10
-0.5
Figure 3. Same as Figure
0
0.5
1.0
1.5
2.0
Relative Error 2 except 10s R I Avalues are used for the
fundamental Q(L,M). of these, RIA, however, appeared to provide good accuracy. The effect of 10s energy interpretation for CH3CN-He is shown in Figure 1. Rates out of the lowest level, R(O,O -+j,k), each of which corresponds to a single QO,k), are compared with C S values. Results are analogous to those found for the linear rotor case: most importantly RIA appears to provide a reasonable approximation. Within the 10s approximation the entire matrix of rate constants is obtained by scaling the R(0,O - + j , k ) . Again, a variety of choices can be made for the 10s energy. Only two will be considered here, corresponding to R I Dand RIA. The original derivation of the ECS scaling6suggests using the former. As might be expected, however, this tends to underestimate all the RO,k -j',k') just as it does the R(0,O -+j,k). If the relative error is defined as relative error = (scaling rate - C S rate)/CS rate the average relative error is -47%; i.e., the scaling predictions are on average a factor of 2 smaller than the CS rates. Figure 2 shows the distribution of relative errors for the 209 state-to-state rates for which C S values are available. Since RIA provides a better approximation to the R(0,O - j , k ) , using it in the 10s scaling relationships is expected to be better. Although this gives an average relative error of + 13% which might
The Journal of Physical Chemistry, Vol. 89,No. 24, 1985
5292
Green
,
2.0
1.0 -
-I 0
I 1 1 1 1 1 1 1
d3
I
I 1 1 1 1 1 1
I
I
I
I 1 1 1 1 1 1
IO+
lo-'2
CS
Rate.
I
1
, ,,
I
l '
, , , , , ,,
, , ,,
, ( , I
, ,
I
/
%
"
,, ~~
-l
I I I
10-10
em's-
Figure 4. Relative errors in 10s predictions of CS state-to-state rate constants as a function of the CS rates. Input Q(L,M) were taken from 10s R I Avalues. Transitions within k ladders are indicated by circles and transitions across k ladders by crosses. 2.0k X
I
I
1.81.6
15
I
I
I
I
I
-
1.4-
-
a ~
1.2-
1.0-
2
1
x
X
x x x
.6-x
--
a X
L
x
d = K
-.6
i I -.8
- 1.0
-8
-.4
0 4 8 Relative E r r o r
12
1.6
20
10
20
30
40
50
60
70
A E , cm-l
-
Figure 5. Distribution of relative errors in the 10s scaling predictions for the CS state-to-staterate constants, using the CS RG,k 0,O) rates as input. The bottom panel shows transitions within k ladders, and the
upper panel transitions across k ladders. appear to be an improvement, the spread of relative errors is now much greater (see Fig. 3) and the root mean square (rms) relative error of 50% is virtually unchanged from the value obtained using the RIDrates as input. Examining separately transitions in which k does not change and those in which it does, it is seen that the rms error in the former decreases from 42% to 33% on going from RID to RIA while in the latter it increases from 58% to 65%. In an attempt to find some systematic basis for the observed relative errors, their correlation with the collisional energy change and with the CS rate were examined, with only limited success. For example, the relative error using RIAas input is plotted as a function of CS rate in Figure 4. For those transitions in which k does not change, the relative error and especially the scatter in the relative errors is seen to increase for the smaller rate constants. Considering that the rates vary by 3 orders of magnitude, the 10s approximation has reasonable predictive power, especially for the larger, more important rates. It is possible that some of the apparent error in the smaller rates is an artifact due to poor convergence or other errors in the CS values. For transitions in which k changes, however, the situation is not so sanguine. These rates vary by only about a factor of 5, so the relative errors of typically 50% are more serious. Furthermore, there do not appear to be any trends in these errors as a function of C S rate or of transition energy. Besides predictions entirely within the 10s formalism, as considered above, it is possible to examine independently the accuracy of the 10s scaling relationships, Le., the ability to predict the entire matrix of CS rate constants given the rates into (or out of) the lowest level. Following the suggestion of DePristo et aL6 the downward CS rates, R(j,k O,O), were used to predict other downward transitions; that is, rates into the lowest level were converted to Q(j,k) via eq 4b and the latter were used in eq 3 to predict other downward rates. Because CS values for R(j,3
-
-
Figure 7. Same as Figure 6 , except that the relative errors are plotted as a function of the inelasticity.
0,O) were available only for j < 7 , these were augmented with 1 0 s R I Apredictions through j = 15. The resulting distribution of relative errors is shown in Figure 5. The statistics here are somewhat better than for the pure 10s methods. First, the distributions tend to be centered more nearly on zero, rather than being systematically high or low. Further, the rms relative error for transitions in which k does not change decreases significantly to 20%, although for transitions in which k does change, the value of 59% is not an improvement. This last statistic, however, is strongly influenced by a few rates that are predicted to be more than a factor of 2 too large, and on closer examination, these involve levels near the basis set limit, where the CS value may be poorly converged, and where use of 10s RIAvalues as input rather than CS rate constants was necessary. Figures 6 and 7 show the relative errors as a function of CS rate and of inelasticity, respectively. For transitions that do not change k, Figure 6 shows that the larger rate constants are given to generally within &IO% by the 10s scaling. The larger rates in this case correspond to smaller inelasticities and analogous behavior is seen in Figure 7 . For transitions in which k changes, the errors are larger and less systematic. That the largest errors occur for small inelasticity may, at first sight, seem strange, but these correspond to large changes in j and involve transitions to levels near the basis set limit; as discussed above the errors here may reflect problems other than the 10s scaling relationship. Because of the 10s energy ambiguity, alternate procedures are also possible. One that was considered was to use the upward CS R(0,O - j , k ) to predict other upward transitions; that is, rates out of the lowest level were converted to QG,k) via eq 4a and the latter were used in eq 3 to predict other upward transitions. Predictions of the 10s scaling using upward transitions were somewhat less accurate than those using doward transitions. In particular, the rms relative error increased from 20% to 25% for transitions in which k does not change and from 59% to 65% for transitions in which it does. It was noticed, however, that errors
The Journal of Physical Chemistry, Vol. 89, No. 24, 1985 5293
Low-Energy CH3CN-He Collisions with these two methods tended to be in opposite directions, suggesting taking an average. Both geometric and arithmetic averages were tried and both provided a significant improvement, especially for transitions in which k changes. The arithmetic average gave the best statistics with rms relative error of 20% for transitions in which k does not change, 24% for transitions in which it does change, and 22% for all 189 transitions for which comparisons could be made. In order to get a more detailed picture of accuracy it is worthwhile to consider in addition to statistical measures the error in individual predicted rates. This will be done here only for 10s scaling with C S downward rates as input, one of the most accurate procedures as judged by overall statistics. For transitions in which k does not change, the behavior is quite similar for k = 0, 1 , and 2: predictions are almost all within f10% except those involving the highest few rotational levels considered, for which the predictions are almost always too low. This might indicate a breakdown of the 10s scaling for transitions with large rotational energy changes. On the other hand, it might also reflect poorly converged CS values. For the k = 0 ladder, it seemed possible that part of the error came from ignoring QG,O) with j > 15 which should contribute to some of these rates; however, the k = 1 and k = 2 ladders show identical behavior despite the fact that none of the required QG,O) are missing. For the k = 3 ladder, only a few rates are available for comparison, but all are predicted quite well. For transitions in which k changes, the behavior appears more complex. For the downward k = 3 0 transitions the 10s scaling is generally accurate to better than about 20% except for rates to levels with j values near the basis set limit, which are predicted to be too large. These predictions often involve Q(j,3) values taken from 10s RIA calculations rather than from CS values as the latter are not available f o r j > 7, and these errors may very well reflect R I Avalues that are too large. It is also possible that C S values for these cases are not well converged. The downward k = 0 3 rates all start from high j levels and all are predicted to be too large, as in the analogous k = 3 0 transitions just described and presumably for the same reason, that the R I , values 1 rates are generally are too large. The downward k = 2 predicted to be about 20-30% too large with no trends apparent. It should be noted that some of these predictions rely on 10s R I A values for Q 0 , 3 ) , and one would expect, based on the above discussion that if the latter are too large, these predictions should be systematically larger than is found here. Finally, only a few 2 rates are available for comparison; these downward k = 1 1 case, mostly rely on 10s RIA values, and unlike the k = 2 predicted values are generally too large as would be expected if the RIA values are too large. The picture that emerges from the above discussion is fairly encouraging. For transitions within a ladder, in which k does not change and which are quite analogous to those in a linear molecule, rates are generally given quite accurately by the 10s scaling. The largest errors occur for highly inelastic transitions with smaller, less important rates, and some of the apparent error here may be due to poorly converged C S rates. For transitions across k ladders, many of the rates are accurately predicted, and in those cases where predictions appear poor, it is quite possible that the error can be attributed to poorly converged C S values or to lack of accurate QG,3) for higher j . The last question to be addressed here is whether 10s scaling predictions can be improved by use of ECS corrections. Because the He projectile is light, one expects the sudden approximation to be relatively good and the ECS corrections to be small. Transitions in which k does not change are formally the same as linear rotor transitions and ECS corrections have been found to be useful for such systems.’ For such transitions in the present system, and with reasonable values for the “critical impact parameter”, I,, of 1-2 A, the ECS corrections were minor and of mixed value in improving the scaling predictions. For transitions in which k changes, however, the ECS correction (in the form originally proposed in ref 6) in many cases was significantly worse than no correction. In particular, for downward k = 3 0 and k =2 1 transitions ECS scaling predicted rates that were too
-
-
-
-
-
-
-
-
-
large by more than a factor of 2. On the other hand, for downward k = 1 2 transitions, for which the 10s scaling was generally poor, the ECS correction often gave significant improvement. For downward k = 0 3 transitions, the ECS correction generally made little difference except that rates predicted to be much too large by the 10s scaling were predicted to be even larger by the ECS correction. A key factor in the ECS correction is choosing an appropriate “energy gap” for each transition. For linear molecules, which have a simple rotational energy level scheme, the method suggested by DePristo et aL6 may work fortuitously well since it captures the essential fact that the spacing between adjacent levels grows linearly with the rotational quantum number. When applied to symmetric tops, however, their scheme becomes complicated, depending on the direction in which both the j and k quantum numbers change; for transitions in which k changes, it appears to be often inappropriate. This problem was noted by Richard and DePristo” in their study of NH3-He, and they suggested a somewhat different scheme for choosing the energy gap in that case and also developed a second-order correction. Several alternative schemes were tested here on the CH3CN-He rates by modifying the energy gap, Q , for cases where k # k’, although no attempt was made to go beyond first-order corrections. In particular, the following were tried:
-
QG,kJ‘,k’) = E(j,3) - E(j - 1,0)
(5a)
Q(j,kJ’,k’) = E(j,k) - E(j’,k’) Q(j,kJ’,k’) = ( 1 / 2 ) E ( j , k ) Q(j,kJ’,k’)= E(j,k) - E(j - l , k - 3 ) Q(j,kj’,k’)= E(j,k) - E 6 - 1,k’) Of these, only the first, eq 5a, was an improvement over the original6,” definition. When it was used, ECS corrections for transitions in which k changes became qualitatively similar to those in which k does not change; that is, corrections were all small and of mixed value in improving the 10s scaling predictions.
Conclusions Large C S scattering calculations were done for low-energy CH3CN-He collisions using a theoretical intermolecular potential. State-to-state rates were obtained for a kinetic temperature of 40 K. Rates among the lower levels are thought to be accurate, but there is some question about basis set convergence for rates to the higher levels. Nonetheless, these are probably the most extensive and accurate rates available for a symmetric top system, and they have been used to test the validity of the 10s approximation and the 10s and ECS scaling laws. Results are fairly encouraging. The 10s approximation provides a reasonable estimate of the fundamental R(0,O j , k ) ; as with linear rotors, a proper intrepretation of the 10s energy is important for highly inelastic transitions, and the “average” values, RIA as defined in ref 16, appear to be a good choice for symmetric tops as well as for linear rotors. Furthermore, the 10s scaling law then provides a fairly good estimate for the entire matrix of state-to-state rates, use of the actual C S values for the fundamental rates being somewhat superior to use of 10s predictions as might be expected. Rates within a k ladder, Le., those in which k does not change, agree with C S values for the most part to within 4=20%. Note that rates within the A and E symmetry species are predicted with equal accuracy although only the former are required as input. Predictions for rates across k ladders do not appear to be predicted quite as accurately. Unfortunately, uncertainties in the accuracy of CS values make it difficult to pinpoint with any certainty whether the error lies with the scaling, the use of 10s values for some of the fundamental rates, or errors in the C S rates. The sudden approximation is expected to be reasonable for this system, suggesting that ECS corrections will be small. For
-
(17) Richard, A. M.; DePristo, A.
E. Chem. Phys. 1982, 69, 273.
J . Phys. Chem. 1985, 89, 5294-5302
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transitions in which k does not change this was found to be true, although the corrections did not generally give improved agreement with the C S values. For transitions across k ladders, on the other hand, the ECS corrections using the formalism of DePristo et a1.6,17 were generally inappropriate. This appears to be due to a poor choice for the energy gap, Q , for these cases. Part of the difficulty here may stem from the symmetry requirement that changes in k must be multiples of three. Several alternate choices were tried, one of which, eq 5a, seemed to be an improvement, although, as for transitions within a k ladder, the corrections for this system while small were of mixed success in improving agreement with CS rates. For systems with larger reduced mass, where ECS corrections are expected to be more important, it will be necessary to consider this point carefully. In concluding, it appears from this study that the 10s approximation and especially the IOS scaling law give an estimate for rates in this system which are probably of acceptable accuracy for many applications including the analysis of radioastronomical
data. Nonetheless, this study suffered from the inability to obtain accurate CS rates among all the levels accessible at even the low kinetic temperature of 40 K, suggesting that alternate methods must be considered. In particular, exact classical trajectories might be useful although a difficulty could arise from the symmetry restriction imposed on changes in k . Also, an approximation was recently introduced by Clary'* that is designed specifically for a system like this; basically it treats only the j quantum number within the IOS approximation while retaining a C S description for the k quantum number. Registry No. He, 7440-59-7; CH,CN, 75-05-8.
Supplementary Material Available: Details of the intermolecular potential used in the scattering calculations (3 pages). Ordering information is given on any current masthead page. (18) Clary, D. C. J . Chem. Phys. 1984, 81, 4466.
An Interpretation of the Sensitivity of Weakly Acidic (Basic) Polyelectrolyte (Cross-Linked and Linear) Equilibria to Excess Neutral Salt Jacob A. Marinsky Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214 (Received: April 23, 1985; I n Final Form: July 16, 1985)
The physicochemical properties of weakly acidic (basic) polymeric gels dispersed in aqueous media have been examined in this manuscript. Two kinds of behavior have been shown to characterize these substances. In the first kind the polymeric macromolecule is permeable to simple neutral salt (hydrophilic) and a Donnan potential term has to be taken into account to understand fully the experimental results. In the second kind the macromolecule is impermeable to the salt (hydrophobic) and the co-ion of the salt medium is not involved in the potential term that has to be taken into account to facilitate interpretation of the experimental results. The most exciting aspect of this research evolves, however, in its extension to weakly acidic (basic) linear polyelectrolytes. The concepts developed and experimentally verified with the various gel systems examined have been shown to apply fully to the linear polyelectrolyte systems as well.
Introduction The extreme sensitivity of the protonation properties of most weakly acidic (basic) gels to the concentration of simple neutral electrolyte during their neutralization with standard base (acid) is a direct consequence of a Donnan potential term arising from the gel's permeability to simple salt.' Correction for this term, accessible from the Donnan-based model. leads to resolution of intrinsic pK values for the repeating monomer unit of these gels which are in agreement with the literature values reported for their monomer counterparts to provide unambiguous proof of the model's validity. The apparent pK (PJ~'&,~ = pH - log a / ( 1 a ) ) of these gels, when rigid, has, in this study, also been found, as predicted by the model, to be a unique function of (pH pX) where X represents the salt medium co-ion activity. In weakly acidic (basic) gels that are impermeable to simple electrolyte the co-ion of the simple electrolyte remains uninvolved in defining the potential term a t the gel-solution interface and the apparent pK of these kinds of gels is expected to be independent of pX. Their apparent pK does indeed become a unique function of the experimental pH alone when the gel is rigid, as predicted.
+
(1) Marinsky, J. A.; Slota, P. "An Electrochemical Method for the Determination of the Effective Volume of Charged Polymers in Solution"; Eisenberg, A., Ed.: American Chemical Society: Washington, DC, 1980; Adc. Chem. Ser. No. 187, 311. (2) Marinsky. J. A.: Lim, F. A,: Chung. K . S. J . Phys. Chem. 1983. 87. 3139.
0022-3654/85/2089-5294$01.50/0
The above analysis of the protonation properties of weakly acidic (basic) salt-permeable and salt-impermeable gels has been presented in detail in this manuscript. Every fundamentally based projection has been verified experimentally. The same two-phase model has been applied as well to the analysis of the protonation equilibria of their linear polyelectrolyte analogues. On the basis of Manning's research3-* the likelihood of its applicability to these systems was considered to be sizeable. Theoretical Treatment of Gels A Donnan-Based Interpretation of the Sensitivity of the Potentiometric Properties of Weakly Acidic Gels to the Presence of Neutral Salt.1,2.9-'2Fundamental Concepts. At equilibrium, during each step of the potentiometric titration of a weakly acidic (3) Manning, G. S. J. Chem. Phys. 1969, 51, 924. (4) Manning, G. S. "Polyelectrolytes"; SElCgny, E., Ed.; Reidel: Dordrecht,
Netherlands, 1974; pp 9-37. (5) Manning, G. S. Annu. Rev. Phys. Chem. 1972, 23, 1 1 7. (6) Manning, G. S. Q. Rev. Biophys. 1978, 11, 179. (7) Manning, G. S. Acc. Chem. Res. 1979, 12, 443. (8) Friedman, R.; Manning, G. S. Biopolymers 1984, 23, 2671. (9) Marinsky, J. A.; Wolf, A.; Bunzl, K. Talanta 1980, 27, 461. (10) Marinsky, J. A,; Gupta, S.; Schindler, P. J . Colloid Interface Sci. 1982,89, 1102, 401. (11) Marinsky, J. A,; Merle, Y. Talanta 1984, 31, 199. (12) Alegret, S.; Marinsky, .I. A.; Escalas, M. T. Talanra 1984, 31, 683.
0 1985 American Chemical Society
