Quantum mechanical calculation of resonance ... - cchem.berkeley.edu

3 downloads 34 Views 184KB Size Report
Jul 1, 1998 - Timothy C. Germanna) and William H. Miller. Department of .... vibrational dynamics in the deep acetylene potential wells themselves, and only ...
JOURNAL OF CHEMICAL PHYSICS

VOLUME 109, NUMBER 1

1 JULY 1998

Quantum mechanical calculation of resonance tunneling in acetylene isomerization via the vinylidene intermediate Timothy C. Germanna) and William H. Miller Department of Chemistry, University of California, Berkeley, California 94720-1460 and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720

~Received 24 February 1998; accepted 27 March 1998! Microcanonical rate constants for the acetylene isomerization reaction have been computed using the direct cumulative reaction probability methodology of Seideman and Miller @J. Chem. Phys. 96, 4412 ~1992!; 97, 2499 ~1992!# and Manthe and Miller @J. Chem. Phys. 99, 3411 ~1993!#. Two- and three-degree-of-freedom calculations are reported using a normal mode Hamiltonian based on the geometry of the vinylidene intermediate. Due to the vinylidene well, numerous resonances are found in the isomerization rate. Little coupling is found between the CH2 rock reaction coordinate and the other normal modes, so that the resonances are readily assignable as normal mode progressions. Qualitatively similar results are obtained using two different potential energy surfaces, or different reduced dimensionality sets of coordinates. © 1998 American Institute of Physics. @S0021-9606~98!02225-9#

Fig. 1, with oxirene (HCOC8 H8 ) playing the role of vinylidene. Lovejoy and Moore used SEP to excite isotopically labeled ketene to a suitably high and well-resolved energy E, and then were able to determine the microcanonical isomerization rate k(E). They observed resonance structure in k(E) which they attributed to metastable states of oxirene. They modeled their results with a one-dimensional resonance tunneling model, and then Gezelter and Miller10 carried out more detailed ~multidimensional! calculations on a realistic potential energy surface, obtaining good qualitative agreement with the overall resonance structure observed. It occurred to us that analogous resonance tunneling structure should be present in the isomerization of acetylene,

I. INTRODUCTION

One of the most interesting features in the vibrational dynamics of acetylene1,2 is the existence of a local minimum on the lowest S 0 potential energy surface with the vinylidene (H2C5C:) geometry ~cf. Fig. 1!. Ab initio calculations3–7 for the ground electronic surface indicate that the isomerization barrier is ;1 – 5 kcal/mol above the vinylidene minimum, and dynamical calculations4 of the isomerization rate give a lifetime of ;1 ps for the metastable states of vinylidene. The most direct experimental observation of the metastable vibrational states of S 0 vinylidene comes from the photodetachment studies of Lineberger and coworkers, who were able to determine three of its six vibrational frequencies, and estimate that its lifetime is between 0.04 and 0.20 ps.8 Field, Kinsey, and coworkers have also obtained evidence2 of isomerization resonances ~due to the vinylidene well! in some of their high resolution stimulated emission pumping ~SEP! spectra of acetylene.1,2 The purpose of this paper is to present calculations relevant to another way for possibly observing the vinylidene states and their lifetimes, namely via resonance tunneling in acetylene isomerization. This is motivated by experiments of Lovejoy and Moore9 which exhibited such resonance tunneling in ketene isomerization,

~1.2! where C and C8 are as above, and H[ 1 H and H8 [ 2 H[D; i.e., one needs the fully isotopically labeled species D13C [ 12CH to distinguish reactants and products. The experiment would be to excite this isotopically labeled acetylene, e.g., via SEP, to a well-defined energy E in the vicinity of vinylidene and to determine the isomerization rate k(E); resonance structure should appear in the energy dependence of k(E) due to the metastable states of vinylidene. The positions and widths of the resonances will give the energies and lifetimes of the metastable vinylidine states. The isomerization rate k(E) could perhaps be determined in a real time pump-probe spectroscopic experiment, or perhaps more simply by carrying out the SEP experiment at high pressure; e.g., at a nonresonant energy E the excited acetylene would be quickly vibrationally relaxed to its parent isotope, while at a resonance energy the isomerization rate would be faster than the collisional relaxation rate. Here we present calculations of k(E) for reaction ~1.2! to indicate semiquantitatively the type of resonance structure that could be observed. Because of the limited accuracy of

~1.1! where C[ C and C8 [ C. The potential along the reaction coordinate for this reaction is qualitatively the same as in 12

13

Present address: Theoretical Division ~T-11, MS B262!, Los Alamos National Laboratory, Los Alamos, NM 87545.

a!

0021-9606/98/109(1)/94/8/$15.00

94

© 1998 American Institute of Physics

Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

J. Chem. Phys., Vol. 109, No. 1, 1 July 1998

T. C. Germann and W. H. Miller

N~ E !5

(k p k~ E ! .

95

~1.7!

The eigenvalues $ p k % ’s have values between 0 and 1, and have an appealing interpretation as the transmission coefficients for the eigenstates of an ‘‘activated complex,’’ in the language of transition state theory. Section II gives more specifics of the computational methods used here, including the reduced dimensionality normal mode Hamiltonians chosen for the present isomerization reaction. Section III presents results for several different sets of reduced dimensionality calculations, and Section IV concludes. FIG. 1. Sketch of the S 0 potential energy surface for acetylene isomerization. Note that the barrier and wells are not to scale; the barrier is ;125 kcal/mol above the vinylidene minimum, and the acetylene minimum ;40 kcal/mol below it. Dotted lines represent negative imaginary absorbing potentials, as described in the text.

the potential energy surfaces and the reduced dimensionality of the calculations, we do not believe the calculated resonance structure to be accurate in detail, but the overall features, e.g., density of structure, typical widths, etc., should be correct. The method of calculation is essentially the same as before:10 the microcanonical isomerization rate is given by k ~ E ! 5 @ 2 p \ r ~ E !# 21 N ~ E ! ,

~1.3!

where r (E) is the density of reactant ~here, acetylene! states per unit energy ~which is a very smooth function of E) and N(E) is the cumulative reaction probability ~CRP! which contains the resonance structure. Seideman and Miller11 showed that N(E) can be expressed as ˆ ~ E ! † eˆ p G ˆ ~ E ! eˆ r # , N ~ E ! 54Tr@ G

~1.4!

where eˆ r and eˆ p are absorbing potentials12 in the reactant and product channels, respectively ~indicated by the broken lines in Fig. 1!, and the outgoing Green’s function is related to the ˆ by total molecular Hamiltonian H ˆ ~ E ! 5 ~ E1i eˆ 2H ˆ ! 21 , G

~1.5!

with eˆ 5 eˆ r 1 eˆ p . By absorbing flux just outside the barriers, as indicated in Fig. 1, one avoids having to describe the vibrational dynamics in the deep acetylene potential wells themselves, and only has to describe the dynamics related to transmission through the double barrier region. The Hamiltonian operator is expressed in a discrete variable representation,13 in which the potential energy terms ~including eˆ ) have simple diagonal matrix representations. The crux of the calculation is the matrix inversion required to obtain the Green’s function. More specifically, we used the approach of Manthe and Miller14 to evaluate the trace efficiently: the Lanczos algorithm is used to determine the eigenvalues $ p k (E) % of the Hermitian reaction probability operator †ˆ ˆ ˆ ˆ 1/2 Pˆ ~ E ! 54 eˆ 1/2 r G~ E ! e pG~ E !e r ,

in terms of which the CRP is given by

~1.6!

II. COMPUTATIONAL METHODS A. Potential energy surface

A schematic of the acetylene isomerization reaction pathway is shown in Fig. 1. The acetylene isomers are separated by a large ~.2 eV! barrier, on which a shallow vinylidene well resides. As for ketene isomerization,10 we will consider a microcanonical distribution in one acetylene well ~ignoring the interesting vibrational eigenstate distribution of highly excited acetylene1,2!, and follow the quantum dynamics until escape to the other isomer well. This may be accomplished by placing absorbing boundary strips12 just beyond the two transition states, as shown in Fig. 1. Increasingly accurate ab initio calculations for this reaction have been carried out over the past 20 years, primarily by Schaefer’s group3,4 and more recently by others as well.5–7 The development of a global potential energy surface ~PES! for this reaction has lagged far behind, however, with the two available PES’s relying on input from the early calculations of Dykstra and Schaefer3~a! and Carrington et al.4 Carter, Mills, and Murrell15~a! first proposed a manybody expansion form, including up to four-body terms, for describing the acetylene isomerization reaction. Spectroscopic information about the acetylene region was combined with the ab initio results of Dykstra and Schaefer3~a! in the vinylidene region to fit the potential parameters; however, the resulting PES was not suitable for dynamics calculations since the saddle point energy ~2.530 eV! was more than 0.4 eV higher than the ab initio value, E TS 52.108 eV. Slight modifications to this surface were then proposed by Halonen, Child, and Carter, with particular focus on accurately reproducing acetylene vibrational term values;15~b! we will refer to this modified surface as the Carter PES. Holme and Levine16 ~HL! later proposed a simple analytical potential for use in their classical dynamics studies interpreting the experimental vibrational spectra1 of highly excited ~.3 eV! acetylene. This PES also suffers from several defects, perhaps most significantly that the superposition of C and H atoms is not forbidden.17 The stationary point energies and vinylidene normal mode frequencies for both of these PES’s are compared with experimental8 and some recent high-level ab initio5–7 values in Table I. We note that the Carter PES tends to overbind vinylidene, while both potentials give reasonable vibrational

Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

96

J. Chem. Phys., Vol. 109, No. 1, 1 July 1998

T. C. Germann and W. H. Miller

TABLE I. Stationary point energies and vinylidene normal mode frequencies.a Energies are in eV ~relative to acetylene!, and frequencies in cm21 .

E vin E TS v1 v2 v3 v4 v5 v6

Carter

HL

MP4b

CCSD c

CCSD~T! d

1.713 2.109 3430 1146 2480 782 3248 356

1.957 2.105 3591 1718 1226

2.010 2.095 3140 1652 1216 721 3232 283

1.862

1.957 2.081 3159 1682 1234 769 3228 327

f

3569 132

3171 1689 1259 776 3267 408

expt. e

3025630 1635610 1165610

v 1 5CH symmetric stretch (a 1 ); v 2 5CC stretch (a 1 ); v 3 5CH2 scissors (a 1 ); v 4 5out-of-plane (b 1 ); v 5 5CH asymmetric stretch (b 2 ); v 6 5CH2 rock (b 2 ). b MP4/TZ1P calculations of Ref. 5. c CCSD/TZ12P calculations of Ref. 6. d CCSD~T!/cc-pVTZ calculations of Ref. 7. e Photoelectron spectra of Ref. 8. f The HL PES is restricted to planar geometries, so the out-of-plane normal mode is missing. a

frequencies ~the glaring exception being the CH2 scissors frequency v 3 for the Carter PES, which is too large by a factor of 2!. FIG. 2. Vinylidene normal mode coordinates, calculated using the Carter PES.

B. Normal mode Hamiltonian and adiabatic approximations

In terms of the mass-weighted normal coordinates at the vinylidene geometry, the molecular Hamiltonian for total angular momentum J50 is taken to be 6

ˆ5 H

(

j51

2

1 ]2 1V ~ Q! , 2 ] Q 2j

~2.1!

i.e., vibrational angular momentum terms have been neglected. The normal mode displacement coordinates Q j for the Carter PES are shown in Fig. 2. Although exact calculations with as many as F56 degrees of freedom are now becoming possible for some systems ~in particular the H2 1OH and HO1CO reactions!,18 the approximate nature of the PES employed here does not justify such expensive calculations. Instead, we will seek to treat the most important few ( f 52 or 3! degrees of freedom in an exact manner, considering the remaining F2 f degrees of freedom as adiabatically separable. These remaining degrees of freedom are then folded in by microcanonical convolution, `

N~ E !5

( N f ~ E2 e nF2 f ! , n50

~2.2!

where e nF2 f denotes the energy of the F2 f independent harmonic oscillator degrees of freedom with quantum numbers n5 $ n f 11 ,n f 12 , . . . ,n F % , F

e nF2 f 5

( \v j j5 f 11

S D n j1

1 . 2

~2.3!

For our reduced dimensionality Hamiltonian, we take the potential energy minimized over all remaining degrees of freedom,

V f ~ Q 1 ,Q 2 , . . . ,Q f ! 5

min

~2.4!

V ~ Q! .

Q f 11 ,Q f 12 , . . . ,Q F

~Simply taking the f -dimensional slice of V(Q… with Q f 11 5Q f 12 5•••5Q F 50 does not, in general, give acetylene minima.! We note that although the $ Q f 11 ,Q f 12 , . . . ,Q F % coordinates are allowed to vary in Eq. ~2.4!, the microcanonical convolution of Eq. ~2.3! uses the normal mode frequencies at the vinylidene geometry Q50. One could go a step further and include the variation of these frequencies with position as well, as Bowman has suggested in treating the bending motions of four-atom systems.19 Rather than performing a single dynamical calculation and then folding in the remaining degrees of freedom by the convolution of Eq. ~2.3!, one would carry out the dynamics for each set of quantum numbers n on the effective adiabatic potential surface V nf 5

min Q f 11 ,Q f 12 , . . . ,Q F

F

F

V ~ Q! 1

( \ v j ~ Q! j5 f 11

S DG n j1

1 2

. ~2.5!

Here we write v j (Q) to indicate that these frequencies are no longer restricted to the vinylidene geometry. However, we found this approach to be unsuitable because of the large adiabatic frequencies v j (Q), which dominate the small energy differences of the isomerization pathway. This is not the case in previous applications,19 where the v j (Q) corresponded to low-frequency bending modes which vanish in the reactant and/or product channels.

Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

J. Chem. Phys., Vol. 109, No. 1, 1 July 1998

T. C. Germann and W. H. Miller

C. Numerical methods

We employ a direct product grid of discrete variable representation ~DVR! points13 as our basis set. For the ‘‘important’’ degrees of freedom, particularly the CH2 rock reaction coordinate, we use a sinc-function DVR,20 in which the grid points are equally spaced. This spacing is conveniently defined in terms of the number of grid points per shortest possible de Broglie wavelength, N B 54 to 6 in the present calculations. The less important ‘‘spectator’’ degrees of freedom may be represented by a Gauss-Hermite DVR,13 where the grid points are laid out according to a harmonic reference potential. For calculations on serial workstations, we then truncate this direct product grid by keeping only those points with potential energies smaller than a particular cutoff (V cut 52E is typically chosen!. On massively parallel computers, it proves convenient to leave the direct product grid untruncated, and use a spatial decomposition to partition wave function ‘‘vectors’’ among processors.21 Communication between processors may be reduced by replacing the sincfunction DVR kinetic operators with a more local low-order ~5- or 7-point! finite difference approximation, which requires only a slight increase in the density of grid points. We use a simple polynomial absorbing potential,

e ~ r ! 5a

S

r2r 0 r max2r 0

D

n

,

~2.6!

in both of the acetylene well entrances. Due to the steep dropoff of the potential as it approaches the acetylene well~s!, the absorbing potential should also rise quickly; hence, we take n54 to 8. ~It has recently been suggested for general systems that this potential dropoff be mimicked by the use of complex absorbing potentials.22! This dropoff also permits a rather narrow absorbing region, with r max2r0 50.2 found to be sufficient. The few nonzero eigenvalues p k of the reaction probability operator @Eq. ~1.6!# are conveniently obtained by the Lanczos algorithm, in which the operator Pˆ (E) is repeatedly applied to an arbitrary initial vector ~with explicit orthogonalization at each iteration! to generate a Krylov basis. The number of such iterations required is approximately equal to the number of nonzero p k , and is between 2 and 6 for the results reported here. Each operation of Pˆ (E) requires two Green’s function operations, which are equivalent to the solution of Ax5b, with A5E1i e2H @see Eq. ~1.5!#. To solve the complex symmetric matrix equation Ax5b, we use the quasi-minimal residual ~QMR! algorithm.23,24 Like the generalized minimal residual method ~GMRES!,25 QMR is an iterative Krylov space method, requiring several ~hundreds to thousands! of applications of the sparse matrix A onto a vector, but QMR was explicitly designed for complex symmetric matrices A, unlike the more general GMRES. More importantly, we have found for the present system that QMR is about six times faster than GMRES, when using a diagonal preconditioner ¯ 5 @ diag~ A!# 21/2A@ diag~ A!# 21/2 A

97

TABLE II. Stationary points on the Carter PES in vinylidene normal mode coordinates. All structures are planar, so the out-of-plane coordinate Q 4 50 in all cases.

Vinylidene Saddle point Acetylene

Q1

Q2

Q3

Q5

Q6

0 0.7505 2.1106

0 20.1030 20.2817

0 0.1476 0.2941

0 60.1368 61.3571

0 61.5448 62.0175

in both cases. This is in accord with comparisons for other reactions which showed that QMR converges faster ~in terms of CPU time! than GMRES, especially when high accuracy is desired.24 On parallel computers, QMR has an even greater advantage since it avoids the communications bottleneck that GMRES has, namely the explicit orthogonalization which takes place each Krylov iteration. III. RESULTS

To determine which normal mode coordinates should be included, in Table II we give the geometries ~in normal mode coordinates! of the vinylidene and acetylene minima, as well as the saddle points separating them, using the Carter PES. All structures are planar, so we can immediately remove the out-of-plane normal mode coordinate Q 4 from further consideration. The most important coordinates are seen to be the symmetric stretch (Q 1 ) and CH2 rock (Q 6 ) modes. Although the asymmetric stretch coordinate Q 5 also changes a great deal during the isomerization reaction, most of this change occurs between the saddle point and acetylene minimum, which is relatively unimportant for the overall dynamics. The CC stretch motion Q 2 is also relatively unimportant, involving a slight shortening of the vinylidene C5C double bond in going to the acetylene triple bond. Finally, the CH2 scissors motion Q 3 is also relatively small, but might be expected to be strongly coupled with the other bending degree of freedom, Q 6 . Thus we will consider two sets of 2D calculations, coupling Q 6 with either Q 1 or Q 3 , and also a 3D calculation including all three normal mode coordinates. To examine the effect of the particular potential surface chosen, we will also perform one set of 2D calculations using the HL PES. Since the density of states r (E) in Eq. ~1.3! is assumed to be a smooth function of energy, both N(E) and k(E) will exhibit similar resonance patterns, so we will just show the calculated CRPs. A. 2D „CH2 rock, CH2 scissors… reduced dimensionality calculations

The 2D (Q 3 ,Q 6 ) potential surface of Eq. ~2.4! is shown in Fig. 3. We observe that the reaction coordinate corresponds to the CH2 rock mode Q 6 , which is nearly completely uncoupled from the CH2 scissors motion Q 3 . Thus we may anticipate a staircase structure in N 2D (E) corresponding to the opening of successive n 3 50,1,2, . . . channels, with steps approximately every v 3 .0.307 eV. Superimposed on this staircase structure will be resonance peaks spaced by v 6 .0.044 eV, corresponding to quantization along the reaction coordinate in the vinylidene well.

Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

98

J. Chem. Phys., Vol. 109, No. 1, 1 July 1998

FIG. 3. Potential energy surface V 2D for the CH2 rocking and CH2 scissors motions. Contour lines are in steps of 0.1 eV, from 1.0 to 2.2 eV. Acetylene and vinylidene wells are marked by A and V, respectively.

Indeed, this is the structure that is found in the computed N 2D (E), shown in Fig. 4. From the preceding discussion, we can readily assign (n 3 ,n 6 ) quantum numbers to the resonance peaks, as indicated in Fig. 4 ~assuming that all resonances are resolved!. We also show the total N(E) calculated by the microcanonical convolution of Eq. ~2.3!. Due to the limited grid of energy points used in this convolution, several extremely narrow resonances are not resolved. ~This is of no great concern, since they are also unlikely to be resolved in any experiments.!

B. 2D „CH2 rock, CH symmetric stretch… reduced dimensionality calculations

The 2D (Q 1 ,Q 6 ) potential surface of Eq. ~2.4! is shown in Fig. 5. Here we observe significant coupling between the

FIG. 4. Cumulative reaction probability N 2D(E) for the CH2 scissors and rocking modes ~solid curve!, and full N(E) obtained by microcanonical convolution ~dashed curve!. The quantum numbers (n 3 ,n 6 ) are shown for several resonance peaks.

T. C. Germann and W. H. Miller

FIG. 5. Potential energy surface V 2D for the CH symmetric stretch and CH2 rocking motions. Contour lines are in steps of 0.1 eV, from 1.0 to 2.2 eV. Acetylene and vinylidene wells are marked by A and V, respectively.

two degrees of freedom, and may expect that the clean identification of the calculated N 2D (E) found in the previous subsection is no longer possible. However, as seen in Fig. 6, we once again have a steady progression of CH2 rock resonances with spacing approximately equal to v 6 , superimposed on a staircase with spacing v 1 .0.425 eV. In fact, the resonance widths in this case are much narrower than in the previous subsection; from harmonic predictions of their locations we estimate that there are two or three extremely sharp resonances (,1027 eV linewidths! unresolved at lower energies. This difference in linewidths may be partly due to the curved reaction path in the present case, but a more important factor is that the barrier in Fig. 5 is roughly 0.2 eV higher than that in Fig. 3. The total N(E) obtained by convolution is also shown in Fig. 6, and looks qualitatively similar to that calculated in the previous subsection, except that the much narrower resonance widths lead to a more slowly rising N(E) with sharper features.

FIG. 6. Cumulative reaction probability N 2D(E) for the CH symmetric stretch and CH2 rocking modes ~solid curve!, and full N(E) obtained by microcanonical convolution ~dashed curve!.

Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

J. Chem. Phys., Vol. 109, No. 1, 1 July 1998

T. C. Germann and W. H. Miller

99

FIG. 9. Cumulative reaction probability N 2D(E) for the CH symmetric stretch and CH2 rocking modes ~solid curve!, and full N(E) obtained by microcanonical convolution ~dashed curve!, using the Holme-Levine PES.

FIG. 7. Potential energy surface V 3D for the CH symmetric stretch and CH2 rocking and scissors motions. The two-dimensional slices in the three panels are for CH2 scissors coordinate Q 3 520.4, 0, and 0.4, from top to bottom. Contour lines are in steps of 0.5 eV, from 1.5 to 5.0 eV.

C. 3D „CH2 rock, CH2 scissors, CH symmetric stretch… reduced dimensionality calculations

Sections of the 3D (Q 1 ,Q 3 ,Q 6 ) potential surface of Eq. ~2.4! with constant CH2 scissors coordinate Q 3 are shown in Fig. 7. The general structure is not surprising, considering the adiabatic 2D surfaces in the previous subsections. The CH2 rock and CH symmetric stretch coordinates are strongly coupled, while the vinylidene well vanishes for large ~positive or negative! values of Q 3 . Thus, we represent the Q 3 coordinate with a Gauss-Hermite DVR of 8–10 points, while a sinc-function DVR is used for Q 1 and Q 6 , as before. The calculated N 3D (E) is shown in Fig. 8, where we

FIG. 8. Cumulative reaction probability N 3D(E) ~solid curve!, and full N(E) obtained by microcanonical convolution ~dashed curve!.

again observe a similar staircase and resonance peak pattern. In this case, the steps occur due to two normal modes (Q 1 and Q 3 ), so a slightly more complicated sequence occurs, but is still readily understood. From the harmonic frequencies, we estimate that there are four resonances ~not shown! at lower energies, which remain unresolved due to their narrow width (,1025 eV! and the cost of such calculations ~;30 min to a few hours for each energy on currentgeneration IBM and SGI workstations!. D. 2D „CH2 rock, CH symmetric stretch… reduced dimensionality calculations using the Holme-Levine PES

Finally, we repeat the 2D calculations of Sec. III B using the HL PES instead of the Carter PES. As mentioned in Sec. II A, the functional forms of these two surfaces differ greatly. The Carter PES arises from a many-body expansion, while the HL PES is more closely related to a Jacobi coordinate treatment, with potential terms depending on the distances and angles of the two H atoms from the CC center of mass. Thus any artifacts in the computed N(E) arising from the defects in these surfaces17 should become apparent by comparing the two sets of N(E) calculations. The adiabatic potential energy surface obtained with the HL PES is qualitatively similar to that shown in Fig. 5, so we will not reproduce it here. However, there is a quantitative difference in normal mode frequencies, as seen in Table I. @Since the HL potential is planar, we must come up with an out-of-plane bending frequency v 4 to use in Eq. ~2.3!. We note in Table I that the ab initio frequencies are all a few percent greater than the experimental frequencies, so we take v 4 5710 cm21 as an estimate of the true frequency.# The calculated N 2D (E) is shown in Fig. 9, along with the full N(E) obtained by microcanonical convolution. The results are qualitatively similar to those in Fig. 6, except that the much smaller v 6 in the present case leads to more closely spaced resonance peaks, and the shallower well depth leads to fewer such peaks. We also see that the resonance peaks only appear for n 1 50; evidently the shallow vi-

Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

100

J. Chem. Phys., Vol. 109, No. 1, 1 July 1998

FIG. 10. Cumulative reaction probabilities for the different sets of calculations, labeled by the PES and normal mode coordinates included in the calculation. Zero-point-corrected energies on the Carter PES for the vinylidene and transition state stationary points are marked by arrows.

T. C. Germann and W. H. Miller

of CH2 rock resonance peaks, to perhaps two or three at most, rather than the six or more seen in our calculations. The inaccurate normal mode frequencies in themselves would not lead to any qualitative changes, merely a modification of the spacings in the N(E) staircase structure. However, if the physical coupling between normal modes is greater than that of the Carter PES, we may expect qualitative differences between the calculated and experimental CRPs. But as seen in Fig. 10, the fundamentally different Carter and HL potentials both give similar CRPs for the 2D (Q 1 ,Q 6 ) calculations, so we expect that future experimental measurements of the acetylene isomerization rate will turn out to have this simple staircase-plus-resonance structure ~with, of course, the ‘‘true’’ normal mode frequencies!, and will provide useful input for the development of a more accurate global potential surface for this system. In conclusion, the present calculations show that there is indeed an observable resonance structure in the energy dependence of the microcanonical rate k(E) for the isomerization H8 C8 [CH↔HC8 [CH8 . Experimental resolution of this structure would give quantitative information about the potential energy surface in the vinylidene region. ACKNOWLEDGMENTS

nylidene well does not lead to resonance structure at energies much above the saddle point. IV. DISCUSSION

The total CRPs obtained in the different calculations are shown together in Fig. 10. We also indicate the Carter PES energies ~including zero-point energy corrections! for the vinylidene and saddle point structures, which correspond to the estimated first resonance peak and reaction threshold, respectively. We see that the broad resonances of the 2D bending (Q 3 ,Q 6 ) calculation lead to a more rapid increase in N(E) with broader features compared to the other calculations. However, including the symmetric stretch coordinate Q 1 in either a 2D or a 3D calculation leads to a more slowly growing N(E) with much narrower ~and thus more difficult to resolve experimentally! resonance structure. The importance of the Q 1 and Q 6 degrees of freedom is anticipated from Table II, which shows that only these two coordinates change appreciably along the reaction path ~and particularly at the critical region near the saddle point!. The similarity of the final results when including Q 3 quantum mechanically or within a microcanonical convolution gives us some confidence that our approximate treatment of the remaining degrees of freedom by convolution will not affect the overall results, although it may of course affect more detailed features such as resonance widths ~e.g., compare Figs. 6 and 8!. Above all else, our calculations are limited by the accuracy of the PES employed. As seen in Table I, the Carter PES greatly overestimates the depth of the vinylidene well, and gives frequencies for the CH2 scissors and CC stretch modes which are greatly different from the experimental ~and ab initio! values. The effect of the ~presumably! much shallower real-world well depth is to reduce the progression

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 No. DE-AC03-76SF00098, and by National Science Foundation Grant No. CHE97-32758. T.C.G. gratefully acknowledges a Research Fellowship from the Miller Institute for Basic Research in Science. ~a! E. Abramson, R. W. Field, D. Imre, K. K. Innes, and J. L. Kinsey, J. Chem. Phys. 80, 2298 ~1984!; ~b! 83, 453 ~1985!; ~c! R. L. Sundberg, E. Abramson, J. L. Kinsey, and R. W. Field, ibid. 83, 466 ~1985!; ~d! J. P. Pique, Y. Chen, R. W. Field, and J. L. Kinsey, Phys. Rev. Lett. 58, 475 ~1987!; ~e! Y. Chen, S. Halle, D. M. Jonas, J. L. Kinsey, and R. W. Field, J. Opt. Soc. Am. B 7, 1805 ~1990!. 2 ~a! Y. Chen, D. M. Jonas, C. E. Hamilton, P. G. Green, J. L. Kinsey, and R. W. Field, Ber. Bunsenges. Phys. Chem. 92, 329 ~1988!; ~b! Y. Chen, D. M. Jonas, J. L. Kinsey, and R. W. Field, J. Chem. Phys. 91, 3976 ~1989!. 3 ~a! C. E. Dykstra and H. F. Schaefer, J. Am. Chem. Soc. 100, 1378 ~1978!; ~b! Y. Osamura, H. F. Schaefer, S. K. Gray, and W. H. Miller, ibid. 103, 1904 ~1981!; ~c! M. M. Gallo, T. P. Hamilton, and H. F. Schaefer, ibid. 112, 8714 ~1990!. 4 T. Carrington, L. M. Hubbard, H. F. Schaefer, and W. H. Miller, J. Chem. Phys. 80, 4347 ~1984!. 5 Ph. Halvick, D. Liotard, and J. C. Rayez, Chem. Phys. 177, 69 ~1993!. 6 ~a! J. F. Stanton, C.-M. Huang, and P. G. Szalay, J. Chem. Phys. 101, 356 ~1994!; ~b! J. F. Stanton and J. Gauss, ibid. 101, 3001 ~1994!. 7 N. Chang, M. Shen, and C. Yu, J. Chem. Phys. 106, 3237 ~1997!. 8 ~a! S. M. Burnett, A. E. Stevens, C. S. Feigerle, and W. C. Lineberger, Chem. Phys. Lett. 100, 124 ~1983!; ~b! K. M. Ervin, J. Ho, and W. C. Lineberger, J. Chem. Phys. 91, 5974 ~1989!. 9 E. R. Lovejoy and C. B. Moore, J. Chem. Phys. 98, 7846 ~1993!. 10 J. D. Gezelter and W. H. Miller, J. Chem. Phys. 103, 7868 ~1995!. 11 ~a! T. Seidman and W. H. Miller, J. Chem. Phys. 96, 4412 ~1992!; ~b! , 97, 2499 ~1992!. 12 ~a! D. Kosloff and R. Kosloff, J. Comput. Phys. 63, 363 ~1986!; ~b! D. Neuhauser and M. Baer, J. Chem. Phys. 91, 4651 ~1989!. 13 J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82, 1400 ~1985!. 14 U. Manthe and W. H. Miller, J. Chem. Phys. 99, 3411 ~1993!. 15 ~a! S. Carter, I. M. Mills, and J. N. Murrell, Mol. Phys. 41, 191 ~1980!; ~b! L. Halonen, M. S. Child, and S. Carter, Mol. Phys. 47, 1097 ~1982!. 1

Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

J. Chem. Phys., Vol. 109, No. 1, 1 July 1998 T. A. Holme and R. D. Levine. Chem. Phys. 131, 169 ~1989!. M. Re´rat, D. Liotard, and J. M. Robine, Theor. Chim. Acta 88, 285 ~1994!. 18 For a recent review see, J. Z. H. Zhang, J. Dai, and W. Zhu, J. Phys. Chem. A 101, 2746 ~1997!. 19 ~a! Q. Sun and J. M. Bowman, Int. J. Quantum Chem., Quantum Chem. Symp. 23, 115 ~1989!; ~b! J. Chem. Phys. 92, 5201 ~1990!; ~c! J. M. Bowman, J. Phys. Chem. 95, 4960 ~1991!.

T. C. Germann and W. H. Miller

101

D. T. Colbert and W. H. Miller, J. Chem. Phys. 96, 1982 ~1992!. T. C. Germann and W. H. Miller, J. Phys. Chem. A 101, 6358 ~1997!. 22 J.-Y. Ge and J. Z. H. Zhang, J. Chem. Phys. 108, 1429 ~1998!. 23 R. W. Freund, SIAM ~Soc. Ind. Appl. Math.! J. Sci. Stat. Comput. 13, 425 ~1992!. 24 H. Karlsson, J. Chem. Phys. 103, 4914 ~1995!. 25 Y. Saad and M. H. Schultz, SIAM ~Soc. Ind. Appl. Math.! J. Sci. Stat. Comput. 7, 856 ~1986!.

16

20

17

21

Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp