Quantum Functional Sensitivity Analysis of the D + H2 Reaction Rate

17740

J. Phys. Chem. 1996, 100, 17740-17755

Quantum Functional Sensitivity Analysis of the D + H2 Reaction Rate Coefficient via the Separable Rotation Approximation Johnny Chang* and Nancy J. Brown Energy and EnVironment DiVision, Ernest Orlando Lawrence Berkeley National Laboratory, UniVersity of California, Berkeley, California 94720 ReceiVed: June 24, 1996X

Sensitivities of the D + H2 cumulative reaction probability (CRP) are calculated for total angular momentum J ) 0 and 6 to test the applicability of the separable rotation approximation. The CRPs and their sensitivities are then thermally averaged to obtain sensitivities of the log-normalized rate coefficient using a single J. This approach has great promise of being a simple, yet accurate, method of providing reliable feedback information to quantum chemists without resorting to full-dimensionality quantal calculations. On the basis of our sensitivity analysis, we propose three potential energy surface (PES) modifications which should remove the remaining discrepancies between the experimental and theoretical rate coefficients, especially at the high temperatures. Interestingly, none of these potential modifications lie at a particularly “high”-energy region of the PES as was previously thought.

I. Introduction A completely rigorous ab initio calculation of quantal rate coefficients is now possible for some of the simplest chemical reactions. The process involves high-level quantum chemistry followed by accurate quantum dynamics. The only inherent approximations are (i) the neglect of nonadiabaticity between electronic and nuclear motion (i.e., the Born-Oppenheimer approximation), (ii) the neglect of electronic and nuclear angular momentum interconversion, and (iii) the assumption that reaction proceeds on only one (usually the ground state) potential energy surface (PES). This calculation is computationally very intensivesone that has been successfully achieved (i.e., corroborated by experiment) only for the H + H2 reaction and some of its isotopomers.1-5 Even then, there are still some discrepancies between experiment and theory, particularly at high temperatures.4,5 Underlying the theoretical road to agreement with experiment is an arduous iterative process of PES refinements and repeated dynamics calculations. It is certainly desirable to have some approximate means of calculating the quantum dynamics, each time on better surfaces obtained from higher level quantum chemistry, before expending a large amount of effort calculating accurate rate coefficients on the best refitted surface. Our interest in this area has been in developing sensitivity analysis tools to provide feedback information for the PES refinement effort. Unfortunately, with approximate dynamics, one can never be completely certain whether the discrepancies with experiment are due to inaccuracies in the PES or due to some inherent approximation(s) in the dynamics calculation. Recently, however, Mielke et al.3,6 presented a new way of calculating rate coefficients based on a separable rotation approximation (SRA), which yields D + H2 reaction rate coefficients that are essentially identical to those of accurate quantal calculations. The agreement between approximate and accurate rate coefficients is better than two significant figures over the entire 167-1500 K temperature range with the largest error (2%) at 167 K. The agreement between theory and the fit7 to experiment7-10 is also quite good (average absolute deviation of only 5%) from 200 to 900 K. But at higher X

Abstract published in AdVance ACS Abstracts, November 1, 1996.

S0022-3654(96)01869-2 CCC: $12.00

temperatures, theory consistently underestimates experiment with the largest discrepancy (30%) at 1500 K. They surmised that a likely explanation for this (barring experimental error) is that some deficiencies remain in the high-energy regions of the PES. Therefore, it seems appropriate to undertake the present sensitivity analysis study using the SRA method to (i) demonstrate that the SRA method is applicable to sensitivities of the cumulative reaction probability, (ii) determine how the PES can be modified to remove the remaining rate coefficient discrepancies, and (iii) provide additional information about the potential-to-rate coefficient relationship for the D + H2 system. The SRA method for calculating rate coefficient sensitivities thus provides a simple, yet accurate, method for providing reliable feedback information to quantum chemists without resorting to full-dimensionality quantal calculations. A description of the methodology including all approximations and details of our calculations is given in section II. The sensitivities of the cumulative reaction probability are presented in section III, followed by the rate coefficient sensitivities and our specific recommendations for potential modifications in section IV. A brief conclusion is given in section V. II. Calculation A. “Approximate” Dynamics. A full-dimensionality quantal calculation of the rate coefficient requires adding contributions from all the partial waves. The separable rotation approximation (SRA) of Mielke et al.3,6 assumes that the cumulative reaction probability11 (CRP) for any two partial waves (J and J′) can be obtained from one another, i.e., rot NJ(E) ) NJ′(E + ∆EJ′J )

(1)

by an energy shift corresponding to the difference in the variational transition state rotational energies of the two partial waves, rot ) EJ′rot - Erot ∆EJ′J J

(2)

† † For a linear rigid rotor, Erot J ) BrotJ(J + 1), where Brot is the rotational constant of the variational transition state. Invoking the separable rotation approximation, the rate coefficient can

© 1996 American Chemical Society

Sensitivity Analysis of the D + H2 Reaction

J. Phys. Chem., Vol. 100, No. 45, 1996 17741

be obtained from the CRP of a single partial wave,

k(T) ) ≈

1

∞

∑(2J + 1)∫0 dE NJ(E)e-E/k T hQJ)0 ∞

B

rot ∞ 1 VTS Qrot (T)e+E J′ /kBT∫0 dE NJ′(E)e-E/kBT (3) hQ

In eq 3, h is Planck’s constant, kB is Boltzmann’s constant, Q is the reactant partition function, and VTS (T) ) ∑(2J + 1)e-E J Qrot

rot

/kBT

(4)

J

is the rotational partition function of the variational transition state (VTS). For a bimolecular A + BC reaction, the reactant partition function is

Q ) φrel(T) QA(T) QBC(T)

(5)

j kBT)3/2/h3 φrel(T) ) (2πµ

(6)

where

is the relative translational partition function per unit volume of A with respect to BC, µ j is the A-BC reduced mass, QA is the internal partition function of A, and QBC is the internal partition function of BC. There are two features of the SRA method that distinguishes it from other J-shifting methods12-16 of its genre, and they account for the method’s remarkable accuracy (at least, as demonstrated for the D + H2 system). One feature is the choice of which single partial wave (J′) to use in the evaluation of k(T) in eq 3. Mielke et al. showed that eq 1 is very accurate at low energies. But at moderate to high energies, the problem of inaccessible quantized transition states, forbidden by angular momentum constraints, presents the single most significant source of error to eq 1. This problem can be eliminated by using a high enough value of J′ (e.g., J′ ) 6) so that no missing quantized transition states are low enough in energy to cause significant errors. Figure 1 shows the J ) 0 and 6 D + H2 CRPs and the corresponding Boltzmann-weighted CRPs (scaled to a maximum of one) at the high- and low-temperature limits.17 These CRPs were calculated using the log-derivative Kohn variational (Y-KVP) method of Manolopoulos et al.18 on the Boothroyd-Keogh-Martin-Peterson19 (BKMP) potential energy surface. It is seen that the problem of inaccessible quantized transition states starts around 0.65 eV for N0(E) as the mismatch between N6(E) and the shifted N0(E) begins to grow. The Boltzmann-weighted CRPs at 167 K magnifies the threshold region of the CRPs, and one can see that shifting N0(E) to obtain N6(E) is a good approximation up to about 0.69 eV on N6(E). The second distinguishing feature of the SRA method is the † (T), use of a temperature-dependent rotational constant, Brot determined from the CRPs of two partial waves. The rationale for this is that when NJ(E) is shifted by an appropriate energy to match NJ′(E) near its threshold, the higher energy regime of NJ′(E) (J′ > J) tends to be slightly overestimated. To correct for this, Mielke et al.3 evaluate the J-specific rate coefficient

kJ(T) )

1 ∞ ∫ dE NJ(E)e-E/kBT hQ 0

(7)

for two values of J (they chose J ) 6 and J′ ) 9), and pick the † value of Brot (T) such that the relation

Figure 1. Cumulative reaction probability of D + H2 with total angular momentum J ) 0 and 6 and Boltzmann-weighted CRPs scaled to a maximum of 1.0 for temperatures T ) 167 and 1500 K. rot

rot

kJ′(T) ) e(-E J′ +E J

)/kBT J

k (T)

(8)

is satisfied exactly for the chosen J and J′. If kJ(T) is evaluated for more than two values of J, then even higher-order expansions of the transition state rotational energy can be used to further improve on the accuracy of the SRA method. Mielke et al.3 discovered yet another approximation that reduces the computational time for the D + H2 CRPs by almost a factor of 2 and which is presumably generally applicable to systems with two identical particles. They found the CRPs for the D + para-H2 and D + ortho-H2 reactions to be almost identical. Thus, only one symmetry block is necessary for a quantum dynamics calculation. Unfortunately, as explained below, both symmetry blocks are needed for a sensitivity analysis. In Table 1, we show some representative D + H2 CRPs for J ) 6 on the BKMP surface for the two symmetry blocks. It is seen that equating the ortho and para CRPs is an excellent approximation except for the very low and high energies. At the reaction threshold, only the ν ) 0, j ) 0 rovibrational state of H2 is open, so only the even symmetry block CRPs will be nonzero. Our calculated D + H2 rate coefficients for the BKMP surface, using both the even and † (T) determined from ref 3, are odd symmetry blocks, and Brot given in Table 2 and compared to those of Mielke et al., who use only the even symmetry block. The agreement is excellent and provides further confirmation that these recent calculations

17742 J. Phys. Chem., Vol. 100, No. 45, 1996

Chang and Brown

TABLE 1: Comparison of the D + Ortho- and Para-H2 Cumulative Reaction Probabilities for J ) 6 on the BKMP Surface E (eV) D + para-H2 D + ortho-H2 E (eV) D + para-H2 D + ortho-H2 0.30 0.33 0.36 0.42 0.45 0.60 0.75 a

1.9(-13)a 2.87(-11) 1.31(-9) 4.73(-7) 5.40(-6) 6.164(-2) 1.605

8.4(-14) 2.75(-11) 1.31(-9) 4.76(-7) 5.41(-6) 6.164(-2) 1.605

0.90 1.05 1.20 1.35 1.50 1.65

5.036 11.32 21.84 36.21 55.69 79.47

5.036 11.33 21.81 36.17 55.78 79.39

δNJ(E) δV(Rγ,rγ,θγ)

)

∑ ∑

(γ)a) (γ′*a) n n′

J δPγ′n′rγn

(13) δV(Rγ,rγ,θγ)

where

Numbers in parentheses indicate powers of 10.

TABLE 2: Comparison of k(T) Calculations for the BKMP Surface using the Separable Rotation Approximation with J ) 6 and a Temperature-Dependent Transition State Moment of Inertia k(T)

(cm3

molecule-1 s-1)

k(T)

(cm3

molecule-1 s-1)

T (K) Mielke et al.a present work T (K) Mielke et al.a present work 167 200 250 300 400 500 600 700 a

use in eq 12. In section IV, we compare log-normalized rate coefficient sensitivities calculated for J ) 0 and J ) 6. For the sensitivity results presented in the next section, the D + H2 CRP sensitivities are given by

3.20(-19)b 3.17(-18) 5.21(-17) 4.20(-16) 6.93(-15) 4.12(-14) 1.42(-13) 3.54(-13)

3.17(-19) 3.16(-18) 5.21(-17) 4.20(-16) 6.92(-15) 4.11(-14) 1.42(-13) 3.54(-13)

800 900 1000 1100 1200 1300 1400 1500

7.23(-13) 1.28(-12) 2.07(-12) 3.10(-12) 4.40(-12) 5.98(-12) 7.86(-12) 1.00(-11)

7.22(-13) 1.28(-12) 2.07(-12) 3.09(-12) 4.39(-12) 5.95(-12) 7.80(-12) 9.93(-12)

Reference 3. b Numbers in parentheses indicate powers of 10.

are better converged than the earlier calculations of Park and Light1b (or that the neglect of Coriolis coupling in their calculations is a serious approximation). The small differences at the high and low temperatures in Table 2 are due to differences in the para and ortho CRPs near the threshold (low T), which we have included, and the contribution from CRPs at energies above 1.68 eV (high T), which we have omitted (see Figure 1). B. Sensitivity Analysis. The SRA method for the CRP sensitivities assumes that eq 1 also holds for functional derivatives with respect to the potential, i.e., rot J′ δNJ(E) δN (E + ∆EJ′J) ) δV δV

(9)

The essential new approximation is that an extra term proportional to δ∆Erot J′J/δV generated from the functional chain rule can be ignored from the right-hand side of eq 9. Using eq 9, we obtain the following functional derivative equivalents of eqs 3 and 7, rot δkJ(T) δk(T) VTS (T)e+E J /kBT ) Qrot δV δV

(10)

J δkJ(T) 1 ∞ δN (E) -E/kBT ) ∫0 dE e δV hQ δV

(11)

and

For the figures shown in section IV, we use partially lognormalized rate coefficient sensitivities J

δN (E) -E/kBT ∞ 1 δk(T) ) ∫0 dE e δV k(T) δV

/∫ dE N (E)e ∞

0

J

-E/kBT

(12)

where the prefactors containing Erot J and the partition functions have been canceled out. Equation 12 should therefore be generally applicable to systems with or without a linear transition state. Again, the only choice to make is what partial wave J to

J δPγ′n′rγn

δV(Rγ,rγ,θγ)

)-

2µ J* S S Im{Sγ′n′rγn ψγn } (14) ∫dΩγ ψγ′n′ p2

J is is a state-to-state transition probability sensitivity,20 Sγ′n′rγn S a scattering matrix element, and ψγn is a flux-normalized scattering wave function. γn is a compound index consisting of the indices for chemical arrangement, vibrational, rotational, and orbital quantum numbers. γ ) a refers to the reactant arrangement. Equation 13 contains a sum of all transition probability sensitivities from each of the open reactant rovibrational states to each of the open product rovibrational states of both product arrangements. The unscaled reactant Jacobi coordinates Rγ, rγ, and θγ correspond to the coordinate of the D atom with respect to the center of mass of H2, the vibrational coordinate of H2, and the angle subtended by Rˆ γ and rˆγ, respectively. In eq 14, µ is the isoinertial reduced mass, and the integration is over the Euler angles, Ωγ ) (Φγ,Θγ,Ξγ), which orient the three-atom system in the lab-fixed frame. For more information on the calculation of transition probability sensitivities, the reader is referred to ref 20. One sees from eq 14 that the essential quantities required for a sensitivity analysis are the scattering wave functions. For systems with two identical particles, such as the present D + H2 system, the additional symmetry allows a decoupling of the system of linear equations into two sets. One set contains even rotational states of the homonuclear diatom and even linear combinations of product states from the two equivalent arrangementssthe so-called para-symmetry blocksand the other contains the odd combinations (ortho-symmetry block). To do a sensitivity analysis, one needs to perform two dynamics calculations, one on each symmetry block, to obtain scattering wave functions corresponding to the unsymmetrized product states. However, once the appropriate unsymmetrized scattering wave functions are obtained, there are two or three additional simplifications one can use to reduce the computation time for sensitivities with essentially no loss in accuracy. Paralleling the similarity in the ortho and para CRPs, we find the sensitivities for the ortho and para CRPs to be almost identical. Figure 2 shows sensitivity maps for the ortho and para CRPs at a total energy of E ) 0.60 eV separated according to odd or even H2 rotational angular momentum quantum number and γ′ ) b or c product arrangement. With the reactant arrangement Jacobi angle θa ) 0°, the steepest-descent path starts out from the bottom of the reactant valley at large Ra (lower right corner in the figures) and ends in the product valley of arrangement γ′ ) b. The portion of the CRP sensitivities for reaction into arrangement c is very small relative to that of b, and the ortho and para contributions tend to cancel each other to a large extent. Furthermore, the small a f c sensitivities lie only in the asymptotic reactant valley. When θa is changed to 180°, the roles played by the product arrangements b and c are switched. Figure 2 suggests that we can ignore contributions from the a f c transition probability sensitivities and need only calculate a f b sensitivities for either the even or the odd rotational states of H2. This approximation reduces the number


J. Phys. Chem., Vol. 100, No. 45, 1996 17743

Figure 2. Sensitivity maps of the J ) 0 ortho and para CRPs at E ) 0.60 eV and θ ) 0° separated into the b and c product arrangements. The contour values are drawn in increments of ∆z shown on the plots. Positive contours are represented by solid lines and negative contours by dashed lines. The zero contour line is omitted. The numbers in parentheses represent the minimum and maximum sensitivity values on an 80 × 65 grid. All sensitivity contour values are given in units of eV-1 bohr-6. Also shown on the plots is the steepest-descent path in unscaled Jacobi coordinates and the location of the barrier top, marked by a heavy dot (b), on the BKMP surface. The incoming reactant arrangement is the lower right asymptotic branch.

of terms that need to be summed (and evaluated!) in eq 13 to about one-fourth of the original. We noticed that the small errors in the CRP sensitivities, introduced by this approximation, all lie in the asymptotic reactant valley (R > 4 bohrs) even at nonzero Jacobi angles and disappear after thermal averaging to obtain the rate coefficient sensitivities. For the few CRP sensitivity slices shown in the next section, we have recalculated the CRP sensitivities without this approximation. A second efficiency found relates to the evaluation of the scattering wave functions at all the coordinates (Rγ,rγ,θγ) where S on a rectangular a sensitivity is desired. The evaluation of ψγn grid is the most time-consuming step in the sensitivity analysis part of our calculations. In the version of the Y-KVP scattering formalism we used (developed by Manolopoulos et al.18b), the wave functions are expanded as linear combinations of basis functions from all three arrangements. Basis functions from arrangement c make no contributions to sensitivity maps when θa e 40° and, therefore, do not need to be evaluated in the sensitivity analysis. (Similarly, basis functions from arrangement b do not contribute to sensitivity maps when θa g 140°.) A third simplification applies when θa ) 0° or 180°. All the dynamics (and sensitivity) calculations are decoupled according to (inversion) parity. The parity is even or odd depending on whether P ) (-1)j+l is +1 or -1. j and l are the rotational and orbital angular momentum quantum numbers, respectively. When θa ) 0° or 180°, only one parity block (j + l + J ) even number) contributes nonzero sensitivity. This is because the angular dependence of the scattering wave function,20 spanned by the Arthurs and Dalgarno coupled angular momentum eigenfunctions in the Euler and body-fixed angle

coordinates,21

Y jlJM(θγ,Φγ,Θγ,Ξγ) ) 2l+1 1/2 J (Φγ,Θγ,Ξγ) (15) ∑k 〈jlk0|jlJk〉Yjk(θγ,0) DkM 4π

( )

vanishes as a result of either the spherical harmonic, Yjk(θγ,0), being zero, for θγ ) 0° or 180° and k * 0,22 or the vector coupling Clebsch-Gordan coefficient, 〈jl00|jlJ0〉, being zero J is a Wigner when j + l + J is an odd number.23 In eq 15, DkM D function in the “consistent passive” convention of Pack and Parker.21c We conclude this section by introducing a short-hand notation for the approximations and efficiencies discussed here. All the sensitivity maps shown in the next two sections include nuclear spin degeneracy. Para-H2 has nuclear spin I ) 0 and a degeneracy of 1 while ortho-H2 has nuclear spin I ) 1 and degeneracy 3. Thus, the exact CRP sensitivity can be written as

δNJ(E) δV

)

∑ ∑ (2I + 1)

P)(1 I)0,1

δNJPI(E) δV

(16)

where NJPI(E) is the CRP for a particular total angular momentum J, parity P, and nuclear spin I. Since δNJP0/δV is equal to δNJP1/δV to an excellent approximation, the CRP sensitivities used in the rate coefficient sensitivity calculation are obtained via

17744 J. Phys. Chem., Vol. 100, No. 45, 1996

Chang and Brown

Figure 3. Comparisons of the J ) 0 (left column) and J ) 6 (right column) CRP sensitivity maps at E ) 0.54, 0.99, and 1.44 eV of N6(E) with θ ) 0°. The details of the contour lines are the same as in Figure 2.

δNJ(E) δV

)4

∑

JP0 δNafb (E)

δV (without c basis functions when θa e 40°) (17)

P)(1

with only one parity block contributing when θa ) 0°. The a f b subscript in eq 16 means that only transition probabilities from arrangement a to arrangement b are included. For notational convenience, we drop the arrangement label subscripts from the Jacobi coordinates, which, henceforth, all refer to arrangement a. III. Sensitivity of the Cumulative Reaction Probability This section presents an assessment of the separable rotation approximation for CRP sensitivities. We calculated the sensitivity of the D + H2 CRP to the BKMP surface for total angular momenta J ) 0 and 6 over an energy range 0.30 eV e E e 1.68 eV. The temperature-dependent rotational constant, † (T), determined from moment-of-inertia data given in ref 3, Brot varies between 6.861 × 10-4 eV at 167 K and 8.869 × 10-4

eV at 1500 K. The corresponding energy shifts, ∆Erot J′J, between N0(E) and N6(E) vary between 0.0288 and 0.0373 eV at the two temperature limits. For comparing the J ) 0 and J ) 6 CRP sensitivities, we used an intermediate energy shift of 0.0314 eV, which brings N0(E) into alignment with N6(E) at the E ) 0.54 eV slice of N6(E). Specifically, N0(0.5086 eV) ) N6(0.54 eV) ) 9.34 × 10-3. All the sensitivity maps shown here are drawn with respect to the unscaled Jacobi coordinates of the reactant arrangement. Because there are three independent variables, R, r, and θ, one needs to hold one of the coordinates fixed in order to plot the contour maps. Results are presented first for θ ) 0° followed by the other Jacobi angles. A. CRP Sensitivities at θ ) 0°. Figure 3 shows a comparison of sensitivities for N0(E) and N6(E) at three energy slices. For the lowest energy slice, the agreement between the sensitivities of N0(0.5086 eV) and of N6(0.54 eV) is excellent. These sensitivities are predominantly negative, indicating that the CRP decreases when the barrier height is slightly increased.


Figure 4. Sensitivity maps of the J ) 6 CRP at E ) 0.69, 0.78, and 0.90 eV with θ ) 0°. The details of the contour lines are the same as in Figure 2.

The largest negative sensitivities are on the barrier shoulders24 (locations which roughly correspond to the classical turning points along the vibrationally adiabatic reaction path). For the comparisons at the higher energies, the agreement is still very good. In fact, it is better than what one might have expected from the rather poignant disagreement in the J ) 0 and 6 CRPs at energies greater than 0.69 eV (see Figure 1). The magnitude of the sensitivities generally increases with increasing energy paralleling the increase in the CRPs. The most interesting feature of these higher energy slices is the large positive sensitivities on the repulsive wall directly facing the entrance channel. The large positive-negative sensitivity dyad indicates that the CRP at these energies can be increased by making the repulsive wall even more repulsive. The question to be answered (in the next section) is whether or not this increase in the CRPs is sufficient to increase the rate coefficients to obtain agreement with experiment at the high temperatures. Another interesting feature of the high-energy slices is the large area of negative sensitivity in the inner corner region.24 This is where

J. Phys. Chem., Vol. 100, No. 45, 1996 17745 the reactive flux “cuts the corner” and a decrease in the potential here will also increase the CRP. Since we are interested in CRP sensitivity features that point to PES modifications which can increase k(T), we present, in Figure 4, three additional energy slices at 0.69, 0.78, and 0.90 eV for J ) 6. These slices lie in the energy regime where the integrand of eq 7 is largest at 1500 K (see Figure 1), and they help to fill the gap in the description of how the CRP sensitivity structure evolves with energy. Starting from threshold, and increasing in energy, we see the large negative sensitivities on the barrier shoulders move toward each other and coalesce to form a single negative sensitivity peak near the barrier top but favoring the reactant side. The positive sensitivity peaks further down the steepest-descent path also move toward each other but grow in magnitude and physical extent. The E ) 0.69 eV slice shows an example of the sensitivity map near the end of this progression. With increasing energy, the positive sensitivity lobe on the product side moves toward the barrier top, pushing the negative sensitivity into the reactant side and into the inner corner region. See the E ) 0.78 eV slice of Figure 4 for an example of this. The magnitude of the sensitivities also reaches a local minimum near this energy. Increasing in energy from 0.78 eV, new positive sensitivity lobes appear on the barrier shoulders and increase in magnitude as they move toward the barrier top. The net effect, as shown by the E ) 0.90 eV slice, is a region of negative sensitivity at the top of the barrier surrounded by three positive sensitivity lobesstwo on the barrier shoulders and one at the outer corner region. Only the positive lobe at the outer corner lies in a high-energy region of the potential. With further increases in energy, the positive sensitivities at the outer corner and on the product shoulder coalesce to form the large region of positive sensitivity at the repulsive wall seen in Figure 3. B. CRP Sensitivities at Nonzero Jacobi Angles. Comparisons of the J ) 0 and 6 CRP sensitivities at nonzero Jacobi angles provide an interesting contrast to the generally excellent agreement seen for θ ) 0°. The separable rotation approximation for CRP sensitivities (eq 9) at nonzero θ is good for energies up to approximately 0.69 eV. This is also the break-off point for the applicability of the SRA to the J ) 0 and 6 CRPs. Above 0.69 eV, the J ) 0 and 6 CRP sensitivity structures evolve in very different ways. Consequently, their sensitivity maps become strikingly different. Figures 5 and 6 show comparisons of J ) 0 and 6 CRP sensitivities at two energiessone below and one above 0.69 eV. For the lower energy slice, the agreement is quite good except for some small differences in sensitivity magnitudes. The overall magnitude of the sensitivities decreases with increasing angle as expected because the minimum-energy path (MEP) is collinear, and wave functions are largest along the MEP when the total energy is low. For the higher energy slice (Figure 6), one sees distinct differences between the J ) 0 and J ) 6 CRP sensitivities. For all regions where there are differences in the sign of the sensitivities, an infinitesimal change in the local potential will affect the J ) 0 and 6 CRPs in opposite ways. The overall magnitudes are also much larger for the J ) 6 CRP sensitivities. The J ) 0 CRP sensitivities decrease in magnitude with increasing angle, but the magnitudes of the J ) 6 CRP sensitivities actually increase in going from θ ) 0° to 10° and to 20° before it starts to decrease. These differences in the J ) 0 and 6 CRP sensitivities are representative of the trends seen at the higher energies and help to explain some of the subtle differences in the rate coefficient sensitivities (next section) determined from the J ) 0 or 6 CRP sensitivities. On the whole, we see that when the SRA holds for the CRPs, it also holds for the CRP sensitivities.

17746 J. Phys. Chem., Vol. 100, No. 45, 1996

Chang and Brown

Figure 5. Comparisons of the J ) 0 and J ) 6 CRP sensitivity maps at E ) 0.63 eV of N6(E) for several values of the Jacobi angle. The details of the contour lines are the same as in Figure 2, except that the steepest-descent path is replaced by the the 1.0 eV equipotentials of the BKMP surface.


J. Phys. Chem., Vol. 100, No. 45, 1996 17747

Figure 6. Comparisons of the J ) 0 and J ) 6 CRP sensitivity maps at E ) 0.78 eV of N6(E) for several values of the Jacobi angle. The details of the contour lines are the same as in Figure 2, except that the steepest-descent path is replaced by the the 1.0 eV equipotentials of the BKMP surface.

IV. Sensitivity of the SRA Rate Coefficient Table 3 shows a comparison of the theoretical and experimental D + H2 reaction rate coefficients. Our theoretical rate

coefficients, calculated on the BKMP surface via the separable rotation approximation, are compared to the SRA k(T) of Mielke et al.3 calculated on the Liu-Siegbahn-Truhlar-Horowitz25

17748 J. Phys. Chem., Vol. 100, No. 45, 1996

Chang and Brown

TABLE 3: Comparisons of k(T) (in cm3 molecule-1 s-1) via the Separable Rotation Approximation on Different Surfaces with Experiment T (K) 167 200 250 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 d

LSTHa d

1.60(-19) 1.74(-18) 3.18(-17) 2.76(-16) 5.04(-15) 3.17(-14) 1.14(-13) 2.92(-13) 6.09(-13) 1.10(-12) 1.79(-12) 2.71(-12) 3.88(-12) 5.32(-12) 7.03(-12) 9.03(-12)

DMBEa

BKMPb

experimentc

1.40(-19) 1.62(-18) 3.12(-17) 2.77(-16) 5.12(-15) 3.24(-14) 1.16(-13) 2.98(-13) 6.20(-13) 1.12(-12) 1.82(-12) 2.76(-12) 3.94(-12) 5.40(-12) 7.13(-12) 9.15(-12)

3.17(-19) 3.16(-18) 5.21(-17) 4.20(-16) 6.92(-15) 4.11(-14) 1.42(-13) 3.54(-13) 7.22(-13) 1.28(-12) 2.07(-12) 3.09(-12) 4.39(-12) 5.95(-12) 7.80(-12) 9.93(-12)

7.25(-20) 1.47(-18) 3.39(-17) 2.96(-16) 5.11(-15) 3.17(-14) 1.15(-13) 3.07(-13) 6.67(-13) 1.26(-12) 2.14(-12) 3.39(-12) 5.04(-12) 7.17(-12) 9.81(-12) 1.30(-11)

a Reference 3. b Present work. c Experimental fit taken from ref 7. Numbers in parentheses indicate powers of 10.

(LSTH) and the double-many-body-expansion26 (DMBE) surfaces. There are several interesting trends in the theoretical results that can be explained by resorting to simple arguments about how differences in barrier heights and barrier widths affect k(T). Comparing just the LSTH and DMBE results, we see that the DMBE rate coefficients are larger than those of LSTH except for the lowest three temperatures. This is because DMBE has a lower classical barrier height than LSTH (9.65 kcal/mol vs 9.81 kcal/mol) and a larger barrier width.20 The lower barrier height accounts for DMBE’s larger k(T) at temperatures above 300 K. For T < 300 K, LSTH’s narrower barrier allows more tunneling and, hence, a larger rate coefficient than DMBE. The BKMP surface has both a lower (9.54 kcal/mol) and narrower barrier20 than both the LSTH and DMBE surfaces. Consequently, its rate coefficients are consistently larger than those of LSTH and DMBE. These speculations on the behavior of k(T) arising from the underlying differences in barrier heights and widths are based on conventional wisdom and will be placed on firmer ground when we present the sensitivities of the rate coefficient. Moving on to comparisons with experiment, we see that the DMBE surface gives the best agreement out of the three surfaces. From 200 to 900 K, DMBE’s k(T) are within ∼10% of the experimental values. BKMP gives considerably larger k(T) for temperatures up to 900 K. We believe that this is mainly attributable to the smaller barrier height and width of BKMP. The most recent high-level ab initio calculations place the barrier height between 9.60 and 9.65 kcal/mol. Partridge et al.27 reported barrier heights of 9.654, 9.649, and 9.632 kcal/ mol using three different basis sets. Diedrich and Anderson28 reported a barrier of 9.61 ( 0.01 kcal/mol using quantum Monte Carlo, and Petersen et al.29 estimate the barrier at 9.60 ( 0.02 kcal/mol. Our conjecture that the BKMP barrier width is too small is based on comparisons with the other two surfaces and the rate coefficient sensitivities presented below. Two more observations before considering the sensitivity maps: All three surfaces tend to overestimate the experimental k(T) at the lowest two temperatures and underestimate k(T) at the highest six temperatures. We shall be looking for sensitivity features which correct for these discrepancies. It goes without saying that the road to agreement between experiment and theory is not unilateral, and we want to mention one area where experiment could benefit from some improvement. The experimental fit of the data points is least satisfactory at the low-temperature limit. Mitchell and Le Roy10 reported a value of 9.60 × 10-20 cm3 molecule-1 s-1 for the rate constant at 167 K, which is

closer to theory than the value from the experimental fit. There are large error bars associated with the experimental measurement10 at this one temperature. Since there have been a number of experimental refinements after the Mitchell and Le Roy experiments were performed, it should now be possible to obtain a more precise determination of k(T) at 167 K. A. Sensitivity of k(T) at θ ) 0°. Figure 7 shows log-normalized rate coefficient sensitivity maps (eq 12) at 200, 400, 1000, and 1400 K calculated on the BKMP surface with total angular momentum J ) 6. The sensitivity maps are superimposed on top of the BKMP surface for easier identification of which regions of the PES need modification. Note that the sensitivity maps are drawn with different contour increments. Our previous experience20,30,31 with transition probability sensitivities on the LSTH, DMBE, and BKMP surfaces indicates that the sensitivities on all three surfaces are very similar, so the information presented here is equally applicable to the LSTH and DMBE surfaces. At 200 K, the regions of largest sensitivity lie on the barrier shoulders. An increase in the potential here will have the greatest negative impact on the tunneling probability and, therefore, the rate. Conversely, if the barrier width is decreased (Viz., the change from DMBE to LSTH), the resultant increase in k(T) could more than compensate for the effects of an increase in barrier height. When the temperature is increased to 400 K, the large negative shoulder sensitivities have coalesced to form a single negative peak with its maximum near the barrier top. The situation vis-a`-vis the effect of changing barrier height versus barrier width on k(T) is now reversed compared to that of 200 K. These observations are compelling evidence for the conjectures presented at the beginning of this section. Another interesting feature of the T g 400 K rate coefficient sensitivities is the growing importance of the positive sensitivities located further down the barrier shoulders. The presence of positive sensitivities for the rate coefficient is a feature that was first seen in our sensitivity study of the collinear H + H2 rate coefficient.32 The ratio of absolute magnitudes of the positive to negative sensitivities grow from 1:10 at 400 K to 1:6 at 1400 K. The location of the positive sensitivities also move up the side of the barrier with increasing temperature. These observations suggest the following potential modification. The potential should be increased in the region near (R, r) ) (2.7, 2.6) bohrs on the product side as well as the equivalent positions in the other arrangements to maintain symmetry. This potential change is tantamount to increasing the width near the lower half of the barrier, and its effect will be to decrease k(T) at 200 K and increase k(T) at 1400 Ksprecisely what is needed to move the theoretical predictions in the direction of experiment. For the intermediate temperatures, there will be some partial cancellation from the positive and negative sensitivities, and the response of k(T) will be more muted. For later reference, we refer to this suggested potential modification as feature i. The overall magnitude of the log-normalized rate coefficient sensitivities decrease with increasing temperature, implying that k(T) is relatively more sensitive to changes in the PES at lower temperatures. The high-energy regions of the PES do not play a significant role in determining k(T) even at 1500 K. The most promising candidates for contributing sensitivity to the highenergy regions of the PES were the large positive sensitivities of the CRP to changes in the repulsive wall seen in Figures 3 and 4, but they do not manifest themselves in the rate coefficient sensitivities until temperatures well above 1500 K. For T g 2000 K, we do see positive sensitivities at the outer corner with magnitudes comparable to those on the barrier shoulder, but these sensitivity maps are not presented here because the upper


J. Phys. Chem., Vol. 100, No. 45, 1996 17749

Figure 7. Sensitivity maps of the log-normalized D + H2 reaction rate coefficient at temperatures 200, 400, 1000, and 1400 K, calculated with J ) 6 and θ ) 0°, superimposed on the BKMP potential energy surface. The sensitivity contour values are drawn in increments of ∆z shown on the plots and are given in units of eV-1 bohr-6. Positive contours are represented by solid lines and negative contours by dashed lines. The zero contour line is omitted. The potential contours, represented by dotted lines, are drawn in increments of 0.4 eV from 0 to 2.0 eV. The numbers in parentheses represent the minimum and maximum sensitivity values on an 80 × 65 grid. The location for the top of the potential barrier is shown by a heavy dot (b).

energy limit (1.68 eV) of our investigation is not high enough to converge the k(T) sensitivities for temperatures above 1500 K. For the “intermediate-energy” regions of the PES, the region between the 0.4 and 0.8 eV equipotential contours, there is considerable negative sensitivity at the inner corner and somewhat less sensitivity at the outer corner. Although the consistent negative sensitivity in these two regions at all temperatures implies that a potential change will move both the high- and the low-temperature k(T) in the same direction, one could still consider increasing the potential at the inner and outer corners to make the passage through the col narrower. This will diminish the low-temperature k(T) significantly more, in a relative sense, than the high-temperature k(T) and at least move ∆k(T)/∆T in the right direction. The decrease in k(T) at high T will need to be corrected for by increases in the barrier width. This potential modification of the inner and outer corners, though of lesser importance, will henceforth be referred to as feature ii. B. Sensitivity of k(T) at Nonzero Jacobi Angles. Figures 8, 9, and 10 show sensitivity maps of the log-normalized D + H2 rate coefficients at nonzero Jacobi angles for T ) 200, 400, and 1400 K, respectively. The magnitude of the sensitivities generally decreases with increasing angle and, like that of the rate sensitivities at θ ) 0°, also decreases with increasing temperature (note the different contour increments). With increasing angle, the regions of large sensitivities move toward the reactant arrangement, although to a lesser extent for T ) 1400 K. This is primarily due to two factors. One, the potential

increases as one moves away from the collinear saddle point with (R, r) fixed, while the relevant energy regime on the potential for determining k(T) remains about the same at a given temperature. Second, at larger angles, the region of the potential past the (noncollinear) barriers leads asymptotically (R f 0) to the H-D-H moiety and not to the reaction products. The most interesting feature of these noncollinear slices, though, is that the region of largest sensitivity changes depending on what temperature is chosen. This is most apparent for the θ ) 30° and 40° slices, where the maximum negative sensitivity peak lies below the 0.4 eV equipotential contour at 200 K, right on the 0.4 eV contour at 400 K, and between the 0.4 and 0.8 eV contours at 1400 K. The changing location of the sensitivity maxima and minima implies that one can change the potential to individually alter only the high- or lowtemperature k(T). For the present purposes, one would like to alter both, and our recommended potential modifications for θ g 30°, denoted collectively as feature iii, are (a) to decrease the potential near (R, r) ) (2.6, 1.65) bohrs so as to increase the high-temperature k(T) and (b) to increase the potential near (R, r) ) (3.2, 1.45) bohrs to decrease the low-temperature (T e 200 K) k(T). These potential modifications should have small-to-negligible effects on k(T) at the intermediate temperatures depending on the relative magnitudes and placements of the PES modifications and the amount of cancellation from the positive and negative sensitivities. C. Comparison of the DMBE-BKMP Potential Difference with the Proposed PES Modifications. It is interesting to compare the difference between the DMBE and BKMP

17750 J. Phys. Chem., Vol. 100, No. 45, 1996

Chang and Brown

Figure 8. Sensitivity maps of the log-normalized D + H2 reaction rate coefficient at T ) 200 K, calculated with J ) 6, superimposed on the BKMP potential energy surface for several values of the Jacobi angle. The details of the contour lines are the same as in Figure 7.



J. Phys. Chem., Vol. 100, No. 45, 1996 17751


Figure 11. Potential difference DMBE-BKMP at various values of the Jacobi angle θ bounded by the 1.0 eV equipotential of BKMP. The contour increments are drawn in increments of ∆z ) 0.05 kcal/mol. The positive and zero contour lines are represented by solid lines and negative contours by dashed lines. The symbols (+) and (×) mark the locations of the PES where the potential should be increased or decreased, respectively.

17752 J. Phys. Chem., Vol. 100, No. 45, 1996 surfaces with the PES modifications suggested by our sensitivity analysis. Figure 11 shows the θ ) 0°, 20°, 30°, and 40° slices of the potential difference DMBE-BKMP bounded by the 1.0 eV equipotential of BKMP. The symbols (+) and (×) mark the locations on the PES where we believe the potential should be either increased or decreased, respectively. For the collinear (θ ) 0°) slice, we have already mentioned that the barrier height and width of the BKMP surface need to be increased. The location of the BKMP saddle point is therefore marked with a (+) near the 0.10 kcal/mol contour line. The (+) symbols encircled by the 0.20 kcal/mol contour lines on the barrier shoulders mark the locations of the PES modifications embodied by feature i. The locations for potential increases suggested by feature ii near the inner and outer corners are also shown in the 0° slice. It is seen that all these PES refinements are already satisfied to a large extent by the DMBE surface, but one might still consider further increases in the barrier width and the potential near the outer corner. The DMBE-BKMP potential difference for θ ) 10° is very similar to the θ ) 0° slice and is not shown here. For potential differences at angles θ g 20°, we have marked the locations of the negative sensitivity extrema for the k(200 K) and k(1400 K) sensitivities. As discussed in connection with feature iii, the potential should be increased at the k(200 K) sensitivity extremum and decreased at the k(1400 K) sensitivity extremum. The DMBE surface satisfies the former requirement but not the latter. D. Sensitivity of k(T) from J ) 0 CRP Sensitivities. Figures 12-15 show log-normalized rate coefficient sensitivities corresponding to those of Figures 7-10, except that these were calculated from sensitivities of the J ) 0 CRP. The salient feature of these J ) 0 sensitivities is that they are, in qualitative terms, essentially the same as those computed with J ) 6. This is due to the fact that sensitivities near the threshold/low energy regions of the CRP, where the SRA is still applicable for J ) 0, account for a substantial part of the k(T) sensitivities. At 1400 K, where CRP sensitivities for E > 0.69 eV constitute the major portion of the k(T) sensitivities, there are minor differences between the k(T) sensitivities computed with J ) 0 or J ) 6sthe largest negative sensitivity of the former lies at a slightly lower energy than that of the latter. There are, however, substantial differences in magnitudes between k(T) sensitivities calculated with J ) 0 or J ) 6. For the θ ) 0° slices, Figures 7 and 12, the sensitivity magnitudes are larger for J ) 0, particularly at higher temperatures. This is because when J ) 0, the inaccessible quantized transition states cause the NJ(E) profile to be much smaller than what would be predicted by a higher J, thus giving too low an estimate for the denominator of eq 12. For the nonzero Jacobi angle slices, the magnitudes of the k(T) sensitivities calculated with J ) 0 and 6 are comparable for T ) 200 and 400 K, but at 1400 K, the small J ) 0 CRP sensitivities cause the numerator of eq 12 to be too small. There is actually some cancellation of errors from both the numerator and denominator being too small when T ) 1400 K. For the high temperatures, the underestimation of the denominator is more serious when θ ) 10° or 20°, while the underestimation of the numerator is more serious when θ ) 30° or 40°. The key point to note, though, is that the k(T) sensitivities with J ) 0 give usefully correct sensitivity information and may be sufficient in many circumstances. This is significant because J ) 0 calculations entail substantially less computational effort. It will still be necessary to do an accurate dynamics calculation because the J ) 0 dynamics results can be very poor, but the shortcut can be applied to the sensitivities.

Chang and Brown V. Concluding Remarks One goal of this paper was to assess the applicability of the separable rotation approximation to calculating sensitivities of the D + H2 cumulative reaction probability. We calculated CRP sensitivities for J ) 0 and 6 and found that, with an appropriate shift in energy, the sensitivities calculated with one J mapped quite well to the sensitivities calculated with the other J up to an energy of ∼0.69 eV on the N6(E) curve. This is also approximately the break-off point for the applicability of the SRA to the two CRPs as well. Above 0.69 eV, we find the CRP sensitivities calculated with J ) 0 and J ) 6 continue to match reasonably well for the zero Jacobi angle, but not for the nonzero Jacobi angles. The evolution of sensitivity structure becomes progressively different with increasing energy when the Jacobi angle moves away from 0°. In particular, the magnitudes of the sensitivities calculated with J ) 0 rapidly diminish with increasing angle, while those calculated with J ) 6 actually increase with increasing Jacobi angle before they eventually decrease. Another goal of this study was to calculate the rate coefficient sensitivities via the SRA to determine how the potential energy surface may be modified to eliminate the remaining discrepancies between experimental and theoretical rate coefficients. We proposed three PES modifications embodied by features i, ii, and iii. These include increasing the width of the barrier, lowering the potential near the barrier for noncollinear geometries, and, of lesser importance, increasing the potential at the inner and outer corner regions. The main surprise is that none of the suggested PES modifications, except for perhaps feature ii, lie at what can be considered a high-energy region of the potential. We believe that the experiment vs theory discrepancies in k(T), at temperatures between 1000 and 1500 K, are most likely due to errors in regions of the potential which lie at energies below 0.6 eV. The suggested PES modifications will, of course, still need to be confirmed by quantum chemistry before one can confer the “ab initio” title on the rate coefficients calculated with the modified PES. Another interesting result of this study is that rate coefficient sensitivities calculated with J ) 0 are in qualitative agreement with those calculated using J ) 6. This means that a J ) 0 sensitivity calculation can still give usefully correct information about how one should modify the potential. The magnitude of the sensitivities, though, can be drastically different between the k(T) sensitivities calculated with J ) 0 or J ) 6 just as the SRA rate coefficients themselves are different depending on whether one chooses J ) 0 or 6 in the calculation. The one calculation that we have not yet performed, mainly because it is beyond the capabilities of our current code and computing resources, is to compute the exact rate coefficient sensitivities by adding the CRP sensitivities from all the relevant partial waves. When such a calculation is performed in the future, it is very likely that it will agree with our current rate coefficient sensitivities calculated with J ) 6 to within 1% or 2% just as the SRA rate coefficients themselves agree with the exact calculation3 to the same precision. Another issue that needs to be addressed is the question of how the inclusion of the geometric phase would affect the results presented in this paper. Wu and Kuppermann33 showed that the geometric phase is important for correctly predicting product rotational distributions and integral cross sections at energies above 1.8 eV, although its effects appear at lower energies for less averaged quantities such as differential cross sections.33,34 Since the rate coefficient is a highly averaged quantity, it is unlikely that the geometric phase will play an important role in the present calculations.


J. Phys. Chem., Vol. 100, No. 45, 1996 17753

Figure 12. Sensitivity maps of the log-normalized D + H2 reaction rate coefficient at temperatures 200, 400, 1000, and 1400 K, calculated with J ) 0 and θ ) 0°, superimposed on the BKMP potential energy surface. The details of the contour lines are the same as in Figure 7.


17754 J. Phys. Chem., Vol. 100, No. 45, 1996

Chang and Brown



Sensitivity Analysis of the D + H2 Reaction Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy, under Contract DE-AC03-76SF00098. References and Notes (1) Park, T. J.; Light, J. C. J. Chem. Phys. 1989, 91, 974. (b) Ibid. 1991, 94, 2946. (c) Ibid. 1992, 96, 8853. (2) (a) Zhang, J. Z. H.; Miller, W. H. J. Chem. Phys. 1989, 91, 1528. (b) Auerbach, S. M.; Miller, W. H. Ibid. 1994, 100, 1103. (3) Mielke, S. L.; Lynch, G. C.; Truhlar, D. G.; Schwenke, D. W. J. Phys. Chem. 1994, 98, 8000. (4) Michael, J. V.; Fisher, J. R.; Bowman, J. M.; Sun, Q. Science 1990, 249, 269. (5) For a recent review of the experiment-theory comparison, see: Michael, J. V.; Lim, K. P. Annu. ReV. Phys. Chem. 1993, 44, 429. For a review of earlier work, see: Truhlar, D. G.; Wyatt, R. E. Annu. ReV. Phys. Chem. 1976, 27, 1. (6) Mielke, S. L.; Lynch, G. C.; Truhlar, D. G.; Schwenke, D. W. Chem. Phys. Lett. 1993, 216, 441. (7) Michael, J. V.; Fisher, J. R. J. Phys. Chem. 1990, 94, 3318. (8) Ridley, B. A.; Schulz, W. R.; Le Roy, D. J. J. Chem. Phys. 1966, 44, 3344. Ridley, B. A. Ph.D. Thesis, University of Toronto, Toronto, Ontario, Canada, 1968. (9) Westenberg, A. A.; de Haas, N. J. Chem. Phys. 1967, 47, 1393. (10) Mitchell, D. N.; Le Roy, D. J. J. Chem. Phys. 1973, 58, 3449. (11) Miller, W. H. J. Chem. Phys. 1975, 62, 1899. (12) For reviews, see: (a) Bowman, J. M. AdV. Chem. Phys. 1985, 61, 115. (b) Bowman, J. M.; Wagner, A. F. In The Theory of Chemical Reaction Dynamics; Clary, D. C., Ed.; Reidel: Dordrecht, 1986; pp 47-76. (c) Bowman, J. M. J. Phys. Chem. 1991, 95, 4960. (13) (a) Sun, Q.; Bowman, J. M.; Schatz, G. C.; Sharp, J. R.; Connor, J. N. L. J. Chem. Phys. 1990, 92, 1677. (b) Schatz, G. C.; Sokolovski, D.; Connor, J. N. L. Ibid. 1991, 94, 4311. (14) (a) Sun, Q.; Bowman, J. M. J. Phys. Chem. 1990, 94, 718. (b) Wang, D.; Bowman, J. M. J. Chem. Phys. 1993, 98, 6235. (c) Wang, D.; Bowman, J. M. J. Phys. Chem. 1994, 98, 7994. (15) Takada, S.; Ohsaki, A.; Nakamura, H. J. Chem. Phys. 1992, 96, 339. (16) The SRA is also similar to the modified wavenumber approximation

J. Phys. Chem., Vol. 100, No. 45, 1996 17755 of atomic physics. For this perspective, see: Takayanagi, K. Prog. Theor. Phys. 1952, 8, 497. (17) Throughout this paper we refer to high/low energy or high/low temperature, which are arbitrary terms. In view of Figure 1 and the temperature range investigated, we shall refer to total energies below 0.4 eV as low energy and above 0.8 eV as high energy. Similarly, temperatures less than or equal to 200 K are low temperatures, and those above 1000 K are high temperatures. (18) (a) Manolopoulos, D. E.; D’Mello, M.; Wyatt, R. E. J. Chem. Phys. 1989, 91, 6096. (b) Ibid. 1990, 93, 403. (19) Boothroyd, A. I.; Keogh, W. J.; Martin, P. G.; Peterson, M. R. J. Chem. Phys. 1991, 95, 4343. (20) Chang, J.; Brown, N. J. J. Chem. Phys. 1995, 103, 4097. (21) (a) Arthurs, A. M.; Dalgarno, A. Proc. R. Soc. London, Ser. A 1960, 256, 540. (b) Miller, W. H. J. Chem. Phys. 1969, 50, 407. (c) Pack, R. T.; Parker, G. A. Ibid. 1987, 87, 3888. (22) Arfken, G. B. Mathematical Methods for Physicists; Academic Press: New York, 1985. (23) Edmonds, A. R. Angular Momentum in Quantum Mechanics; Princeton University Press: Princeton, NJ, 1974. (24) For semantic purposes, the inner (outer) corner of the potential refers to the region on the upper wall of the saddle point where the D-H-H conformation is less (more) compact than the conformation at the top of the barrier. The shoulder regions refer to the opposite locations about halfway down the barrier along the minimum-energy path. (25) (a) Liu, B. J. Chem. Phys. 1973, 58, 1925. (b) Siegbahn, P.; Liu, B. Ibid. 1978, 68, 2457. (c) Truhlar, D. G.; Horowitz, C. J. Ibid. 1978, 68, 2466; 1979, 71, 1514(E). (26) Varandas, A. J. C.; Brown, F. B.; Mead, C. A.; Truhlar, D. G.; Blais, N. C. J. Chem. Phys. 1987, 86, 6258. (27) Partridge, H.; Bauchlicher, Jr., C. W.; Stallcop, J. R.; Levin, E. J. Chem. Phys. 1993, 99, 5951. (28) (a) Diedrich, D. L.; Anderson, J. B. Science 1992, 258, 786. (b) Diedrich, D. L.; Anderson, J. B. J. Chem. Phys. 1994, 100, 8089. (29) Peterson, K. A.; Woon, D. E.; Dunning, Jr., T. H.; J. Chem. Phys. 1994, 100, 7410. (30) Chang, J.; Brown, N. J.; D’Mello, M.; Wyatt, R. E.; Rabitz, H. J. Chem. Phys. 1992, 97, 6226, 6240. (31) (a) Chang, J.; Brown, N. J. Int. J. Quantum Chem., Quantum Chem. Symp. 1993, 27, 567. (b) Chang, J.; Brown, N. J. Int. J. Quantum Chem. 1994, 51, 53(E). (32) Chang, J.; Brown, N. J.; D’Mello, M.; Wyatt, R. E.; Rabitz, H. J. Chem. Phys. 1992, 96, 3523. (33) (a) Wu, Y.-S. M.; Kuppermann, A. Chem. Phys. Lett. 1993, 201, 178. (b) Kuppermann, A.; Wu, Y.-S. M. Ibid. 1993, 205, 577. (34) Wu, Y.-S. M.; Kuppermann, A.; Lepetit, B. Chem. Phys. Lett. 1991, 186, 319.

JP9618690

Quantum Functional Sensitivity Analysis of the D + H2 Reaction Rate

Recommend Documents