12535
J. Phys. Chem. 1993, 97, 12535-12540
Time-Dependent Quantum Study of the HCN
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HNC Isomerization
Boon Leong Lan and Joel M. Bowman' Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322 Received: June 14, 1993; In Final Form: August 18, 1993a
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We report a rigorous, time-dependent study of the three-dimensional H C N H N C isomerization. Wavepackets are prepared by applying an impulse to three bend-excited stationary states that are localized in the H C N well. The time dependence of the wavepackets is obtained by expansion of the initial wavepacket in a basis of 900 vibrational eigenstates and subsequent propagation in this basis. The vibrational eigenstates were calculated previously using an accurate ab initio potential [Bowman, J. M.; Gazdy, B.; Bentley, J. A.; Lee, T. J.; Dateo, C. E. J . Chem. Phys. 1 9 9 3 , 9 7 , 3 0 8 ] . The time-dependent density to be in the H N C well is calculated for many initial impulses, and time-averaged reaction probabilities are calculated as functions of both the average energy of the full wavepacket and the average energy of the components of the wavepacket that isomerize. The contribution of tunneling to the isomerization is calculated and shown to be mainly due to delocalized states with energies below the ground-state adiabatic barrier to isomerization. The cumulative reaction probability is also calculated and shown to have the expected RRKM steplike structure.
I. Introduction There have been few dynamical studies of isomerization reactions. Those that have been made employed a variety of approximations, ranging from the use of classical mechanics to describe the time evolution to reducing the number of degrees of freedom to one, the reaction coordinate, in approximatequantum treatments. For symmetric isomerizations, the one-dimensional symmetric double well has served as a guiding model. In this model the isomerization in the tunneling region is described by the splittings in the nearly degenerate symmetric and antisymmetric eigenstates of the symmetric double well. There have been a number of quantum, semiclassical, and classical studies of two-mode, symmetric double-well systems. Christoffel and Bowman' calculated the effect of excitation of a vibrational mode transverse to the reaction coordinate on splittings, using perturbation theory, self-consistent-field (SCF) theory, adiabatic theory, and quasiclassical trajectories. The SCF and adiabatic results compared very well to exact calculations, based on self-consistentfield theory with configuration interaction. Farrelly and Smith2 investigated several semiclassical approaches to essentially the same model and obtained very accurate splittings. De Leon and Berne3 investigated isomerization dynamics classically for a twomode symmetric double-well potential. They focused on the conditions for the validity of classical RRKM theory and emphasized the importance of strong mode coupling and "stochasticdynamics" as a condition for itsvalidity. Makri and Mille+ applied SCF and adiabatic methods with a clever choice of basis to accurately describe the tunneling splittings in a two-mode, symmetric double-well model representing proton transfer in malonaldehyde. Very recently, Ram et al.5 considered this system and applied exterior complex scaling to obtain isomerizationrates in the tunneling and nontunneling regions. Nonsymmetric isomerization reactions such as HCN HNC cannot be treated by the splitting method, and other methods, e.g., classical trajectories, RRKM theory and variants of it, and semiclassical methods, have been used to study such isomerization reactions. There have been numerous approximate studies of the HCN H N C isomerization using these methods. Gray et alU6 reported a reaction path Hamiltonian for the isomerization based on ab initio calculations. The isomerization rate was calculated using a vibrationally adiabatic approximation and a one-
-
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* Abstract published in Advance ACS Abstracts,December 1,1993.
dimensional,semiclassical treatment of tunneling. The emphasis of their study was on the importance of tunneling in the reaction, and it was concluded that tunneling is significant up to 8 kcal/ mol below the ground-state adiabatic barrier. A more sophisticated semiclassical study of the isomerization was reported by Waite,' using methods developed by Waite and Miller.8 Holme and Hutchinson9 investigated the effect of nonlinear resonances between the bend and H atom stretch on the isomerization rate using classical trajectories. BaEic et al.1° reported a threedimensional study of tunneling in the HNC isomerization using the semiclassical SCF method and with the three-dimensional Murrell-Carter-Halonen (MCH) potential." Smith et a1.I2 applied quasiclassical adiabatic switching in a two-mode model of the isomerization using the MCH potential. They focused on the importance of bend excitation in inducing chaotic motion. Founargiotakis et aI.13reported a three-dimensionalclassical and two-mode quantum mechanical study of regular and irregular dynamics in HCN using the MCH potential. This study also implicated bend excitation in the irregular dynamics. All of the calculations of the HCN isomerization have applied localized or, equivalently, short-time dynamics to study the isomerization. Such approaches bypass the fact that, quantum mechanically, isomerizationreactions involve a discrete spectrum of bound states. Peric et a l l 4tooknoteof this in their modification of RRKM theory, without tunneling, to determine isomerization rates in HCN using their ab initio potential.ls Unlike the theory of bimolecular reactions, for which a rigorous quantum mechanical theory exists as well as numerous rigorous calculations, no rigorous calculations for nonsymmetric isomerization reactions have been reported. Such calculations require a solution of the exact time-independent or time-dependent Schriidinger equation for energies above and below the barrier to isomerization. Only recently have such calculations been possible. Specifically, a number of calculationshave been reported for the HCN/HNC system which have extended above the barrier to isomerization.16-20 We recently reported such calculations on a new, global, ab initiopotential surface.I9,20 Vibrational energies were compared with all experimental data. For 91 HCN vibrational energies, the median and mean absolute differences with experiment are 17.2 and 23.6 cm-1, and for the seven transitions in HNC, the mean and median absolute differences with experiment are 21 and 16 cm-1. The stimulated e n h i o n
0022-3654/93/209712535%04.00/0 0 1993 American Chemical Society
Lan and Bowman
12536 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993
pumping spectrum was also simulated and compared to experiment, with very good results. In this paper we examine isomerization dynamics in H C N by propagating wavepackets initially localized in the H C N well. The propagation is achieved using a large basis of vibrational eigenstates, calculated on the new ab initio potential surface, as described in detail elsewhere.20 In the next section we describe the methods used to prepare and propagate wavepackets. The analysis of the propagation and a discussion of the results are given in section 111. A summary and conclusions are given in section IV. 11. Theory and Calculations
The coordinates used in the calculations are R, the distance of H to the center of mass of CN, r, the C N internuclear distance, and y, the angle between the Jacobi vectors R and r, such that y equal to zero corresponds to linear H C N and y equal to s corresponds to linear HNC. In our previous calculationsZOwe identified localized H C N and H N C states as well as delocalized states. Localized states are predominantly confined to either the H C N or H N C well, whereas delocalized states extend significantly over both wells. This distinction between localized and delocalized states is clear for nonsymmetric isomerization reactions, and thus the threshold for isomerization can be defined as the energy of the first delocalized state. Since isomerization in H C N / H N C involves mainly bending motion, we focus here on wavepackets which initially have excitation in the bending degree of freedom. The approach we take is to apply an impulse to several highly excited, but localized, H C N bend states with zero total angular momentum and to propagate the resulting nonstationary wavepackets. Specifically, we use the vibrational eigenstates (0,22,0), (0,20,0) and (0,20,1) as templates for wavepackets. The first quantum number in the parentheses refers to the H-CN stretch, the second to the H-CN bend, and the third to the C N stretch. For harmonic systems, applying the factor exp(ikx) to a stationary Gaussian wave function produces a nonstationary state, which semiclassically moves with momentum hk and has an additional kinetic energy equal to hZk2/2p.21922 Taking this cue from harmonic systems, the initial wavepacket in our study is given by
where 4,(x) is one of the three vibrational eigenstates mentioned above and where x represents the three coordinates R, r, and y. The factor exp(iky) serves as the (norm-preserving) impulse term. The variable y in eq 1 could be any of these coordinates; however, we restrict attention here to the bending coordinate, and thus y equals cos y. This impulse model is not meant to represent any specific physical excitation process; however, it could be an approximation for an impulsive, hard-collision model. If the initial wavepacket is expanded in terms of a complete set of vibrational eigenstates, (&(x)}, the formal solution to the time-dependent Schradinger equation is
where
are the expansion coefficients and E, are the vibrational energy eigenvalues corresponding tothe+,(x). Thecoefficients andepend on k , the wavenumber, and parametrically on the vibrational eigenstate index j . However, for simplicity, we do not indicate the dependence on these variables. Because the eigenstates q5j(x) used to construct the initial wavepacket are chosen to be localized H C N wave functions, the wavepacket at t = 0 is also localized.
Thus, the expansion coefficients anare zero or very nearly so if &(x) in eq 2b is an H N C state. Nonzero coefficients are possible only if &(x) is either an H C N state or a delocalized state. Therefore, to a very good approximation, the sum in eq 2a is over H C N and delocalized states only. Also, note that the expectation value of the energy is given as usual by
(3) However, below we shall introduce another average energy which has more physical meaning for the isomerization. Using eq 2a, the total probability density in the H N C well, which serves as our definition of the isomerization probability, is defined by
where the integration is over all R a n d r but from yspto T , where yspis the saddle point value of y, 76'. From eq 4b, it is clear that P H N C ( t ) is an almost periodic functionz3of time, t ; therefore, its long-time average
exists and is given by
Expressions similar to eqs 4a,b and 5a,b can be written for and ( P H C N ( t ) ) wherein the integrals are over the H C N well instead of the H N C well. The densities and their corresponding long-time averages in the two wells are related through PHCN(t)
The integral over the H N C well in eq 4bis significantly different from zero only if &(x) and ~ $ ~ ( are x ) either H N C or delocalized states. For H N C states, however, the expansion coefficients an are very small since the overlap of +(x,O) with H N C states is very small. Thus, only delocalized states contribute significantly to P H N C ( t ) and ( P H N C ( t ) ) . Consequently, if the delocalized states are not significantly populated relative to the H C N states initially, P H N C ( t ) is very nearly zero for all times and thereby ( P H N C ( f ) ) is also very close to zero. This can occur in two ways: when k z 0, a, z 6nJ for any choice of H C N state for I#J,(X)or when the H C N state +j(x) has very little overlap with delocalized states. The second condition is met by H C N states with little if any bend excitation. In order to significantly populate the delocalized states initially, it is evident that the H C N states +j(x) should be ones with substantial bend excitation, because these states have large standard deviations in y and would have good overlap with delocalized states. An example of such a state is the (0,22,0) state shown in Figure 1. This is the highest-energy localized H C N bend state for zero total angular momentum and with zero quanta of stretch excitation. The standard expression for the expectation value of the energy of a wavepacket is given by eq 3. Only localized H C N and delocalized states contribute to the summation, whereas, as noted above, only delocalized states contribute significantly to the isomerization probability. We desire an expression for the average energy of the components of the wavepacket that lead to isomerization. There are several possible ways to define such an
HCN
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H N C Isomerization
The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12531
180.0
1
21000
20000
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'E. 19000
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r
135.0
A
w
h
(r
23
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-
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-
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45.0 16000' 0.0
'
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'
200.0
'
'
'
'
'
500.0
300.0 400.0
'
600.0
k2 (au)
0.0
1.5
2.5
2.0
3.0 R
3.5
4.0
4.5
(bohr)
Figure 2. Average energy, ( E ) ,versus k2 for three sets of wavepackets, built from the HCN vibrational eigenstates indicated.
Figure 1. Contour plot of the HCN eigenfunction (0,22,0) in thevariables R and 7 , with r held fixed at its equilibrium value for HCN. The plot of the wave function is superimposed on a contour plot of the potential surface.
TABLE I: Energies (cm-1) Relative to the HCN Minimum and Bond Lengths (bohrs) for HCN, HNC, and the Saddle Point. Denoted CHNS
0.5 0.4 h
c Y
HCN
CHNr
HNC
energy
0
rCN rCH Q4H
2.196 2.016 4.212
16866 2.220 4.097 1.877
5202 2.258 2.226 2.644
0.3
0
z
I
n
0.2 0.1
0.0
energy. For example, restricting the summation in eq 3 to delocalized statesonly (and renormalizing the probability) would be a reasonable definition which is consistent with the observation that only delocalized states contribute significantly to the isomerization probability. Another expression, and the one we adopt, which is consistent with the expression for the isomerization probability given by eqs 4a and 5b is
This energy is normalized with respect to the time-averaged probability of isomerization and is weighted by the contribution of a given vibrational eigenstate to isomerization. Note also that from eq 1, in the limit k 0, the wavepacket is just the eigenstate 4j(x),andboth ( E )and ( EHNC) equaltheenergyoftheeigenstate, Ei. In this limit, the time-dependent and time-averaged isomerization probability, eqs 4b and 5b,are simply given by the integral J H N C ~ ~ (dx, X )which is just the fraction of the localized HCN vibrational eigenstate probability in the H N C well. This fraction is much less than one and could be considered a tunneling probability, as discussed in detail in the next section.
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111. Results and Discussion
In order to intrepret the results, we have given in Table I the relevant characteristics of the H C N / H N C a b initio potential. As seen, the electronic barrier to isomerization, measured relative to the absolute minimum of the H C N well, is 16 886 cm-I. The ground-state adiabatic barrier is given by the sum of the electronic barrier plus the zero-point energies of the two normal-mode stretches at the saddle point. In the harmonic approximation for the normal-mode stretches, the ground-state adiabatic barrier, E&,, equals 19 134 cm-I. In conventional RRKM theory, without tunneling, the threshold energy for the isomerization is determined by this barrier height. In the following we consider wavepackets given by eq 1, where the +(x) are the three H C N vibrational eigenstates (0,20,0),
L
4
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-0.1
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'
0.2
t
'
I
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0.3
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'
0.4
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0.5
(PSI
Figure 3. PHNC(~) up to 0.5 ps for wavepackets built from the state (0,22,0) with average energies (EHNC) of 17 787, 18 816, 19 000,and 21 793 cm-I. Increasing oscillation amplitude correlates with increasing average energy.
(0,22,0),and (0,20,1). These wavepackets were propagated, for a range of wavenumbers k, according to eq 2a with a basis of 900 vibrational eigenstates. This basis spans a large range of energies, from the zero-point energy of HCN, 3477 cm-1, to 28 924 cm-1. This large basis was necessary to preserve the unitarity of the propagation over the range of wavenumbers necessary to obtain substantial isomerization probabilities. In order to relate the wavenumber k to the average energy, we determined the dispersion curves ( E )and (E H N Cversus ) k , using eq 3 and eq 8, respectively. The dispersion curves for ( E ) versus k2 are shown in Figure 2. The nearly linear dependence on k2,except at the higher energies, is as expected from simple semiclassical arguments, as discussed above. The turnover of ( E )a t the highest value of k* shown for each curve may indicate the limitations of the vibrational eigenstate basis, rather than a significant deviation from the simple semiclassical expectation. As mentioned above, in the limit k 0,both ( E ) and (EHNC) equal the energy eigenvalue E,, which for the states (0,20,0), (0,22,0), and (0,20,1) is 16 586, 17 697, and 18 589 cm-I, respectively. Next we consider the time-dependent and time-averaged isomerization probabilities. A . PHNC(f). The isomerization probability, PHNC(t), given by eq 4b, is shown for short times in Figure 3, for initial wavepackets which are constructed from the (0,22,0) H C N state (see Figure 1) for four values of k , 0.5,3.0,5.0,and 15.0. For this vibrational eigenstate, these values of k correspond to average isomerization energies, ( E H N c )of , 17 787, 18 816, 19 000, and 21 793 cm-', respectively. For the smallest values of ( E H N c ) PHNC(f) , is essentially zero for all times, not surprisingly. For the next two
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Lan and Bowman
12538 The Journal of Physical Chemistry, Vol. 97,No. 48, 1993
1
0.5
0’40
L -
-
0.20 -
-
0.10 -
-
0.30 A L Y
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0.5
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2.0
1.5
2.5
16000
17000
(PSI
18000
20000
19000
Icm-
21000
’
Figure 5. (PHNC(t)) versus (E) for the wavepackets derived from the vibrational eigenstates indicated.
h
0.4
L Y
y
0.3
I
p. 0.2 0.1
0.0
I 1 0.0
2.0
4.0
6.0
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Figure 4. Upper panel is the same as Figure 3 but up to 2.5 ps and PHNC(f) for the 21 793-cm-1 packet up to 12.0 ps (lower panel).
values of (EHNc),P H N C ( f ) exhibits the expected almost-periodic behavior. In contrast, for the largest value of (EHNC) during the same time interval, P H N C ( t ) increases rapidly from its initial value of about zero and fluctuates about a value that is significantly greater than zero; i.e., P H N C ( t ) does not recur close to its initial value. Note also that the first maximum in P H N C ( t ) occurs a t shorter times as ( EHNC) increases. Thesecurves are plotted again in the upper panel of Figure 4 for times up to 2.5 ps, and in the lower panel, P H N C ( f ) for the highest energy is plotted for times up to 12 ps. Since P H N C ( t ) is an almost periodic function of time, it should recur close to zero eventually; however, up to a t least about 60 ps, no recurrence close to the initial value (of zero) was observed for this probability. The shift in the initial-value recurrence to longer time as k increases can be understood qualitatively as follows. Each complex expansion coefficient time evolves according to a,(t) = a, exp(-iE,t)
= r, exp[i(O, - E,+)]
where r, and e,, are the modulus and phase of the initial value a,. For P H N C ( f ) to recur close to its initial value, the phase of each coefficient must also recur sufficiently close to their initialvalues. The number of states M which participate in the wavepacket dynamics depends on k: from our numerical analysis, we know that, loosely speaking, Mincreases with k. Hence, a s k increases, the phase matching required for recurrence becomes less probable, and therefore the recurrence to the initial value is less frequent. It is interesting that the three average energies, (EHNc), for which near recurrence occurs on a short-time scale are below E&,, whereas for the highest-energy case, where ( EHNC) is above thls barrier, recurrence does not occur, at least up to 60 ps. We
16000
I
I
I
18000
20000
22000
I
cm’
24000
’
Figure 6. Semilog plots of (PHNC(f)) versus (EHNC) and the tunneling contribution to (PHNC(f)) for wavepackets built from the vibrational eigenstate (0,20,0).
will give an explanation of this result below when we discuss the energy dependence of the longtime average of P H N C ( f ) , which we consider next. B . ( P H N C ( f ) ) . The long-time average of the isomerization probability, ( P H N C ( t ) ) , is shown in Figure 5 as a function of the average energy (E) for the three sets of wavepackets constructed from the vibrational eigenstates (0,20,0), (0,22,0), and (0,20,1). The behavior of ( P H N C ( t ) ) with increasing average energy is similar for the three states: an initial steep rise followed by, roughly, a plateau. For (0,20,1), there is a second plateau. The use of (E) in these plots illustrates its inappropriateness. The time-averaged probabilities for the wavepackets built from the eigenstates (0,20,0) and (0,22,0) would beexpected to havesimilar threshold energies, close to the ground-state adiabatic barrier, E L . As seen, using the conventional average energy (E), the corresponding threshold energies are not similar, and they are both far below E&,. Similarly, the threshold energy for the wavepackets built from the eigenstate (0,20,1) would be expected to be close to the corresponding adiabatic barrier ,Til of 21 121 cm-l, whereas it too is far below that value. As discussed above, the average energy of the wavepacket, (E),contains contributions from nonisomerizing (HCN states) and isomerizing (delocalized) components of the wavepacket. A more appropriate average energy, it was argued above, is (EHNC), given by eq 8. (Recall that (EHNC) cannot be below the energy of the template Vibrational eigenstate.) We have replotted these three probabilities (along with tunneling contributions, which we discuss below) as functions of (EHNC) in Figures 6-8. The first thing to observe from these plots is that the threshold energies for the three probabilities are essentially the same, Le., all
HCN
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The Journal of Physical Chemisrry, Vol. 97, No. 48, 1993 12539
HNC Isomerization
L
0.60 0.50
loo
0.40 0.30
0.20 0.10
,
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Figure 7. Same as Figure 6 but for wavepacket built from the vibrational eigenstate (0,22,0).
0
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Figure 8. Same as Figure 6 but for the wavepackets built from the vibrational eigenstate (0,20,1).
approximately 18 5OOcm-I. This energy isvery close to the energy of the first delocalized state, which is 18 683 cm-1. On the basis of this "experimental" finding, it appears reasonable to define the lowest threshold energy as the energy of the first delocalized state, which is 45 1 cm-1 below the ground-state adiabatic barrier height. Second, note that the isomerization probabilities for wavepackets built from the (0,20,0)and (0.22,O) vibrational eigenstates are essentially the same, except in the deep tunneling region. This indicates that the isomerization probability is insensitive to the precise form of the initial wavepacket, provided the two stretch quantum numbers are the same. Consider next the energy dependence of (PHNC(f)) for wavepackets built from the (0,20,1) eigenstate. As noted the threshold energy is close to 18 500 cm-I; however, this probability shows a marked steplike structure, with a secondary threshold energy of roughly 21 000 cm-I, which is approximately equal to the adiabatic barrier with one quantum of C N vibration, Le., Eil. Indeed, if the isomerization reaction is vibrationally adiabatic, the threshold energy for these wavepackets should be governed by this barrier. Thus, the observation of a much lower threshold energy indicates substantial nonadiabaticity in the isomerization for the C N vibrationallyexcited. That this is indeed nonadiabaticity and not tunneling through the adiabatic barrier can be verified by noting that the energy of the first delocalized state with one quantum of C N excitation is 20 525 cm-1, which is only about 596 cm-1 below the corresponding adiabatic barrier height. It is of interest to determine the extent of tunneling in the isomerization. For H C N / H N C we found six delocalized states below the ground-state adiabatic barrier to isomerization, for
Figure 9. Cumulative reaction probability (solid curve) versus (EHNc). The dashed curves are the time-averaged probabilities for wavepackets built from the state (0,20,0)(open circles) and from the state (0,20,1) (open squares).
zero total angular momentum. Clearly these delocalized states should be regarded as tunneling states. However, it must be noted that all localized HCN (HNC) states have some tunneling amplitude in the H N C (HCN) well. Thus, it is important to distinguishbetween delocalized tunneling states, which have large amplitude in both wells, and tunneling from states that are localized in one well and which have very small amplitude in the other well. We chose to include both types of states to the tunneling contribution. Thus, the tunneling contribution to (PHNC(t)) is obtained simply by restricting the sum in eq 5b to all molecular eigenstates with energies below the appropriate adiabatic barrier height, E L for wavepackets with zero quanta of stretch excitation initially, and Eil for wavepackets built from the (0,20,1) state. Theresulting tunneling probabilitiesareshown in Figures 6-8. We haveverified that except at the lower energies the tunneling contribution for the wavepackets with no initial excitation in the stretches is dominated by the six delocalized states with energies below E L . The tunneling contribution to (PHNC(f)) for wavepackets built from the eigenstate (0,20,1) was determined using the adiabatic barrier height Eil in restricting the sum in eq 5b. This is in strict keeping with the adiabatic theory of isomerization. However, it is clear from the discussion above that the large "tunneling" contribution to the probability for energies between roughly 19 000 and 20 OOOcmo* should be regarded as mainly nonadiabatic contributions to the probability. At the lowest energy shown, the tunnelingcontribution is simply the fraction of the HCN vibrational eigenstate's density to be in the H N C well. In this limit the isomerization probability (PHNC(f)) [and PHNC(f)] is a constant given by this fraction. While this limit is reasonable mathematically, it could be argued that this constant residual probability should be subtracted from (PHNC(f));doing so would then yield a zero value for the isomerization probability at the lowest value of (EHNc).That is our recommendation, even though we have not done that in the plots shown in Figures 5-8. It is of interest to consider the cumulative reaction probability (CRP). The CRPdivided by thedensity of states0fthe"reactant" HCN is (to within a constant) the microcanonical rate of the reaction. In RRKM theory, the CRP is approximated by the number of open states at the transition state at the total energy E, and thus it is a staircase function of E, with steps of unit height. In a more rigorous theory the CRP the sum of reaction probabilities over initial states. In the present context, and for the energy range of interest, the CRP is the sum of (PHNC(f)) for wavepackets built from (0,20,0) and (0,20,1). We have plotted the CRP versus (EHNC) in Figure 9, and as seen, it has a distinctive RRKM-like staircase structure, with "risers" appearing in the vicinity of the thresholds for the ground and first excited states
Lan and Bowman
12540 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 180.0
IV. Summary and Conclusions
135.0
-.. 45.0
0.0
3.0 R (bohr)
2.25
15
3 75
45
Figure 10. Contour plots of the wavepacket at 0.7 ps derived from the ) to 19 000 cm-l. The C N bond length state (0,22,0) for ( E H N Cequal is fixed at 2.25 bohrs. 180.0
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We presented a rigorous time-dependent study of the HCN H N C isomerization. Initial wavepackets were constructed from impulses applied to localized, highly excited bend states of HCN. The wavepacket propagation was achieved by a spectral representation in a basis of 900 vibrational eigenstates which were calculated previously using a new, accurate ab initio potential energy surface. Time-dependent and time-averaged isomerization probabilities were calculated as functions of the conventional average energy of the wavepacket and also as functions of the average energy of the isomerizing components of the wavepacket. The latter energy was shown to be more physically meaningful. Tunneling was examined, and a distinction was made between components of the wavepacket that are delocalized vibrational eigenstates with energies below the adiabatic barrier to isomerization and components of the wavepacket that are localized H C N states which have small tunneling amplitudes in the H N C well. The cumulative reaction probability was also calculated and shown to have the expected RRKM-like staircase structure; however, the magnitude was only about one-third the RRKM result. Contour plots of two wavepackets were presented. These showed, in a preliminary way, a correlation between intermode energy exchange in the H N C well and the presence or lack of recurrence in the time-dependent isomerization probability.
135.0
Acknowledgment. We thank Dr. Bela Gazdy for help with the calculations. Support from the National Science Foundation (CHE-9200434) is also gratefully acknowledged.
h
M
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+
References and Notes
-,
,I -: im-=
0.01 I 1.5
I
I
,
2.25
I
I
I
30 R (bohr)
i
3.75
4.5
Figure 1. Same as Figure 10 but for ( E H N Cequal ) to 21 793 cm-' and 0.3 ps.
of the transition state, Le., E&, and EAl, respectively. However, the magnitude of the CRP is about one-third the RRKM value. Finally, reconsider the recurrence or lack thereof in PHNC(t) seen in Figures 3 and 4 for wavepackets built from the state (0,22,0). As discussed above, the time-averaged reaction probability at energies below E&, is dominated by a handful of delocalized states with energies below E&,. Thus, it is not surprising that PHNC(t) for ( E H N Cequal ) to 18 816 and 19 000 cm-I shows recurrenceon a short timescale. By contrast, PHNC(t) for ( E H N Cequal ) to 21 793 cm-' contains contributions from many vibrational eigenstates, and recurrence does not occur on a picosecond time scale. The lack of recurrence and its relationship to strong intermode coupling were stressed by DeLeon and Bernej in their elegant classical analysis of isomerization. To examine this briefly here, we have plotted in Figures 10 and 11 contour plots of wavepackets built from the (0,22,0) eigenstate with average energies of 19 000 and 21 793 cm-I, respectively. As seen, the wave function for the lower energy is nodeless in the H-CN stretch, but for the higher-energy wave function there is a node in that stretch in the H N C well. This indicates a correlation between short-time recurrence and little if any intermode energy exchange and very long-time recurrence and significant intermode energy exchange between the bend and H-CN stretch in the H N C well, in general accord with the classical analysis of deLeon and Berne.3
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