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A close-coupling study of collision-induced dissociation in He + H2
17 February I995
FIEiAL LETTERS ELSEVIER
Chemical Physics Letters 233 ( 1995) 399-404
A close-coupling study of collision-induced dissociation inHe+H, K. Nobusada a, K. Sakimoto a, K. Onda b aInstitute of Space and Astronautical Science, Yoshinodai, Sagamihara 229, Japan ’ Faculty of Industrial Science and Technology, Science University of Tokyo. Oshamanbe, Hokkaido 049-35, Japan Received 26 September 1994; in final form 2 December 1994
Abstract The previous quantum mechanical close-coupling study of the dissociative collisions in He + Hz (T-shape configuration) by the present authors [Chem. Phys. Letters 216 (1993) 6131 has been extended to a wide range of the total energy 4.8-7.5 eV. The dissociation process shows a clear vibrational enhancement. The dissociation probability has an undulation as a function of the total energy, and by using classical S-matrix theory it is shown that this feature is caused by interference of the contributions of multiple trajectories.
1. Introduction
In atom-diatom collisions at energies above the dissociation threshold, we have a double continuum of scattering and dissociative states. Since it is not easy to solve the quantum mechanical problem of double continuum states, not so many elaborate quantum mechanical calculations have been made for dissociative collisions [ l-121. When a collision system has no rearrangement channel, it is reasonable to regard the dissociation of a molecule merely as a vibrational excitation to continuum states. In this case, the collision process can be described in the framework of the standard close-coupling (CC) method, in which the total wavefunction is expanded in terms of the vibrational eigenfunctions of the target molecule. Diestler and co-workers have first applied the CC method to the dissociation process [ 31. Following their work, the present authors have applied the CC method to the dissociative He + H, collisions on a realistic potential energy surface [ 111 (hereafter 0009-2614/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01462-O
referred to as paper I), and have found that the CC method can accurately describe the dissociation process. In paper I, we have considered He + H2 collisions only at some limited total energies just above the dissociation threshold ( = 4.75 eV) . In this Letter, we have extended the calculation to a wide range of the total energy (4.8-7.5 eV) and have studied the energy dependence of the dissociation probabilities. As in paper I, we have assumed that the He atom approaches along the line bisecting the H2 internuclear distance and all the particles remain in a plane during collisions: i.e. the T-shape configuration. In order to obtain a deeper understanding of the collision dynamics, we have also made classical and semiclassical calculations.
2. Theory We employ the notation as described in paper I. Let distance of HZ, and R be the distance between He and the center-of-mass of H,. The r be the internuclear
400
K. Nobusada et al. /Chemical
Schrijdinger equation for the He + H, system is given by ti2 a2 -----2~ aR2 =Ep(R,
,“,: ,“:l + V(R, r) + V,(r) r) ,
(1)
(-j$+k$1
xn,,n(R) (2)
where kfg is the channel wave number defined by ki, = (2p/h2)(Ea,) with E,, being the vibrational energy of H2 for 0 G n’ Q 14 and being the energy eigenvalue of discretized continuum states for n’ > 15, and (r$,,r 1VI &” ) means the integral over r. The scattering s-matrix elements S,.,n are obtained by imposing the usual boundary condition on x~,,,( R) in the asymptotic region of R + ~0.The probability for the transition from the initial vibrational state n to the final state n ’ is P,,, = ]S, P,n]2.The dissociation probability for the initial state n is given by =
c PI’>15
p,*,,
We define the kinetic energy distribution of the H atom produced by the dissociation of the H, molecule in the initial vibrational state n as
WR, r)
where m and Al.are the reduced masses of H, and of the collision system, respectively; V(R, r) + V,,(r) is the potential energy surface of the T-shape He + H2 system, V,(r) being the potential energy of H,, and E is the total energy measured from the bottom of the H, well. The interaction potential V(R, r) becomes zero as R+m. We have used the analytic fit of the ab initio potential given by Varandas and Brandb [ 131 for V(R, r) + V,(r). The contour map of the potential energy surface is shown in paper I. As in paper I, we discretize the dissociative continuum by assuming that the vibrational eigenfunctions of H2 always vanish outside a box with size r,,. We must take the box size r,, sufficiently large so that the calculated transition probabilities become independent of r_. We set r,, = 15.5 bohr, which is sufficiently large to obtain the accurate transition probabilities reported in this paper. Using the eigenfunctions c#+,( r) of the H2 bound states (0 < n Q 14) and of the discretized H2 dissociative states (n > 15)) we expand the total wav&,(r)X,.,,(R).Then,we efunctionas!P”(R,r)=C,, obtain a set of coupled equations in the form
pdiss ”
Physics Letters 233 (1995) 399-W
.
(3)
(n’> 15) .
(4)
The kinetic energy Ek of the dissociative fragment H is measured in the center-of-mass system of H2, i.e. Ek = 4( E,,,- D), where D is the dissociation energy of the H, molecule ( =4.75 eV). The density of states den,./dn’ has been calculated from the discretized energy levels by using a finite difference formula. Integration of the distribution over the kinetic energy Ek gives the dissociation probability.
3. Results Fig. 1 shows the dissociation probabilities for n = 1, 3,5,7 and 9 in the energy range E = 4.8-7.5 eV. This figure clearly indicates a vibrational enhancement: that is, at a fixed total energy the dissociation probability becomes larger with increasing initial vibrational quantum number. As is often used in other papers, we introduce a loose definition of the dynamical threshold, above which the dissociation probability becomes noticeable. From Fig. 1, we see that the dynamical threshold becomes higher for lower n. This trend in the dissociation probability curve is characteristic of a system that has no rearrangement channel [ 10,14-201. The present work shows that as n increases, the dynamical threshold gets closer to the dissociation threshold. A similar conclusion has also been obtained in classical trajectory calculations [ 18,201. It is interesting that the probability curves for the higher vibrational states shown in Fig. 1 have ‘shoulders’ (or undulation). For a collision system that has no rearrangement channel, the shoulder structure has been also found in semiclassical [ 14,151 and quantum mechanical [ lo] calculations. However, no physical explanation has been given for the presence of the shoulder structure. To find out the physical reason, we have investigated the kinetic energy distribution of the H atom produced by the dissociation.
K. Nobusada et al. /Chemical Physics Letters 233 (1995) 399-404
401
(a) E = 5.62 eV I2
0.01 (b) E = 6.43 eV
i
(cl E = 6.92 eV
“.05 s.0
5.5
6.5
6.0
E
20-
1.5 -
IO-
7.0
:
(ev)
Fig. 1.Dissociation probabilities for n = 1,3,5,7 and 9 as a function of the total energy.
Fig. 2 shows the kinetic energy distribution (4)) as an example, for n =7 at three total energies: (a) E = 5.62 eV (just below the first shoulder position in Fig. 1) ; (b) 6.43 eV (just below the second one) and (c) 6.92 eV (just above the second one). At E = 5.62 eV, only a single peak is present. When E = 6.43 eV, this peak moves towards higher Ek,and another peak appears at lower Ek When E = 6.92eV, these two peaks go to the high energy region and a third peak grows at Ek= 0 eV. Here, we call such peaks peak 1, peak 2 and peak 3 as indicated in Fig. 2. It should be noted that the kinetic energy distribution shows another oscillatory pattern at each time after an additional shoulder appears in the Pfssversus E curve. Therefore, we can expect that the oscillatory pattern in the kinetic energy distribution is closely related to the shoulder structure in the dissociation probability. To make this clear, we show the energy dependence of the area of peak 1, peak 2 and peak 3 in Fig. 3. The sum of the areas over all the peaks is the dissociation probability for n = 7. The area of peak 1 increases with
02
0.4
06
08
1.0
E, WI Fig. 2. Kinetic energy distributions of the dissociative fragment H, dP/dE, versus Ek, for n = 7 at (a) E = 5.62 eV; (b) E = 6.43 eV and (c) E=6.92 eV.
energy up to E = 5.8 eV, and has a constant nearly value of about 0.4 beyond E = 5.8 eV. This makes the first shoulder at 5.7 d E< 6.0 eV, as seen in Fig. 1. As the total energy increases further from E = 6.0 eV, the area of peak 2 becomes appreciable. This area increases and becomes nearly constant ( = 0.1) above E = 6.6 eV. This corresponds to the second shoulder at 6.4 GE < 6.7 eV. Thus we can conclude that the shoulder structure results from the oscillatory pattern in the kinetic energy distribution.
4. Classical and semiclassical calculations To understand the reason for the oscillatory pattern seen in the kinetic energy distribution we have also carried out a calculation using the classical S-matrix theory developed by Miller [ 21,221 and Marcus [ 231.
402
K. Nobusada et al. /Chemical Physics Letters 233 (1995) 399-404
spondingly, two classical trajectories contribute to the 12,-+ n2 transition. Then, the semiclassical transition probability for ni + n2 is calculated by [ 21,221
Peak 1
PE”, = /c
[2di
(!$J”’
0.2
where @( n2, ni) is an action integral along the trajectory and the sum is taken over the two possible values of ql. Owing to the presence of the phase factor @( n2, ni ) , an interference effect can be taken into account in classical S-matrix theory. If we neglect the interference effect, we have the classical transition probability in the form
CG $ .%
0.2
a"
Peak 2
0.1
(6)
0.a
i.0
/ 5.5
I
I
6.0
6.5
E
/
,
7.0
E = 5.62 eV
(ev)
Fig.3.Areasofpeakl(O),peak2(A)andpeak3(W)forn=7 as a function of the total energy. For the definition of peaks 1,2 and 3, see Fig. 2.
(Another kind of semiclassical approach may be applicable to treat this problem [ 241.) Rusinek and Roberts [ 14,151 first applied classical S-matrix theory to the study of dissociation processes. Here, we follow their method. For the description of the dissociative continuum, we have employed the box-discretization method [ 14,151, which is just the same as that used in the present quantum mechanical calculation. We introduce the quantities nl and q, which characterize the classical vibrational motion of H2 before collision: n, is the classical correspondence of the vibrational quantum number and q, is the initial value of the angle variable conjugate to the action. Let n2 be the corresponding quantity after collisions. We must seek classical trajectories which have non-negative integers for ni and n2. This can be achieved by plotting n2 as a function of q, for a given non-negative integer n,. Fig. 4 shows an example for n, = 3, 5 and 7 at E = 5.62 eV. If n2 takes a non-negative integer below its maximum, we always have two roots of ql, Corre-
0.0
0.2
0.4
0.6
0
91/2x Fig. 4. Classical trajectory calculations of the final vibrational quantum number for n, =3, 5 and 7 at E=5.62 eV as a function of the initial angle 4, divided by 2a.
K. Nobusada et al. /Chemical
(a) E = 5.6’2 eV
0.0 20
(b) E = 6.43 eV
O?
403
the oscillatory pattern in the kinetic energy distribution is caused as a result of the interference of the two classical trajectories. This interference effect is just the same as the one pointed out by Miller for the boundbound vibrational transition [ 221. The present work shows that the interference effect is also significant in the bound-continuum transition.
I
*
I
0.1
Physics Letters 233 (1995) 399-404
0.3
0.4
0.5
: : :
5. Conclusion An oscillation is seen in the kinetic energy distribution of the dissociative fragment H. This oscillatory pattern makes a shoulder structure in the probability curve for the dissociation. Using classical S-matrix theory, we have shown that the oscillation in the kinetic energy distribution is due to the interference of two classical trajectories. Therefore, we can conclude that the shoulder structure is an interference effect. The shoulder structure has been seen also in the dissociation of the collinear configuration [ 10,141. Probably, this is an interference effect as well. We can expect that the interference effect in the dissociation is present for all the configurations. It is interesting to see whether the interference effect is observed in 3D calculations.
Fig. 5. Kinetic energy distributions of the dissociative fragment H, dP /dE, versus Ek. for n = 7 calculated by quantum mechanical (O), semiclassical (0) and classical (A) methods at (a) E= 5.62 eV; (b) E=6.43 eV and (c) E=6.92 eV.
The dissociation probability and the kinetic energy distribution are calculated in the same way as in Eqs. (3) and (4)) respectively. Here, we have considered only classically allowed transitions. In the actual calculation of Eq. (5), we have used the uniform approximation
Acknowledgement The authors would like to thank Professor Y. Itikawa for his reading the original manuscript carefully. This work is partially supported by a Grant-in-Aid for Scientific Research on Priority Area “Theory of Chemical Reactions” from the Ministry of Education, Science and Culture of Japan.
[221. Fig. 5 shows the kinetic energy distribution of the dissociative fragment H for n = 7 calculated by semiclassical (open circles) and classical (open triangles) methods. The closed circles are the results of the quantum mechanical calculation shown in Fig. 2. The semiclassical distribution agrees well with the quantum mechanical one except for the high kinetic energy region Ek, in which the transition to these states is classically forbidden. However, the classical method, neglecting the interference effect, fails to reproduce the oscillatory pattern. From these facts, it is evident that
References [ 11 G. Wolken Jr., J. Chem. Phys. 63 (1975) 528. [2] L.W. Ford, D.J. Diestler and A.F. Wagner, J. Chem. Phys. 63 (1975) 2019. [ 31 E.-W. Knapp, D.J. Diestler and Y.-W. Lin, Chem. Phys. Letters 49 (1977) 379; E.-W. Knapp and D.J. Diestler, J. Chem. Phys. 67 ( 1977) 4969. [4] D.J. Diestler, in: Atom-molecule collision theory, ed. R.B. Bernstein (Plenum Press, New York, 1979) p. 655. (51 K.C. Kulander, J. Chem. Phys. 69 (1978) 5064. [61 J.C. Gray, GA. Fraser, DC. Truhlar and KC. Kulander, J. Chem. Phys. 73 (1980) 5726.
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K. Nobusada et al. /Chemical Physics Letters 233 (1995) 399-404
[7] J. Manz and J. RBmelt, Chem. Phys. Letters 77 (1981) 172.
[ 81 J.A. Kaye and A. Kuppermann, Chem. Phys. Letters 78 ( 1981) 546; 115 ( 1985) 158; Chem. Phys. 125 (1988) 279. [9] C. Leforestier, Cl. Bergeron and P.C. Hiberty, Chem. Phys. Letters 84 (1981) 385; C. Leforestier, Chem. Phys. 87 (1984) 241; Chem. Phys. Letters 125 (1986) 373. [lo] Cl. Bergeron, P.C. Hiberty and C. Leforestier, Chem. Phys. 93 (1985) 253. [ 111 K. Nobusada, K. Sakimoto and K. Onda, Chem. Phys. Letters 216 (1993) 613. [ 121 K. Sakimoto and K. Onda, J. Chem. Phys. 100 (1994) 1171; K. Sakimoto, Chem. Phys. Letters 228 (1994) 323. [ 131 A.J.C. Varandas and J. Brandl%o,Mol. Phys. 57 (1986) 387. [ 141 I. Rusinek and R.E. Roberts, J. Chem. Phys. 65 (1976) 872; 68 (1978) 1147.
1151 I. Rusinek, J. Chem. Phys. 72 (1980) 4518. 1161 N.J. Brown and R.J. Munn, J. Chem. Phys. 56 (1972) 1983. ] 171 N.C. Blais and D.G. Truhlar, J. Chem. Phys. 66 (1977) 772. I181 J.E. Dove and S. Raynor, Chem. Phys. 28 (1978) 113. [19] J.E. Dove, M.E. Mandy, N. Sathyamurthy and T. Joseph, Chem. Phys. Letters 127 (1986) 1; J.E. Dove, M.E. Mandy, V. Mohan and N. Sathyamurthy, J. Chem. Phys. 92 ( 1990) 7373. [20] T. Lehrand J.W. Birks, J. Chem. Phys. 70 (1979) 4843. [21] W.H. Miller, J. Chem. Phys. 53 (1970) 1949. [22] W.H. Miller, J. Chem. Phys. 53 (1970) 3578. 1231 R.A. Marcus, Chem. Phys. Letters 7 (1970) 525. 1241 D.G. Trnhlar and J.W. Duff, Chem. Phys. Letters 36 (1975) 551; D.G. Truhlar, Intern. J. QuantumChem. Symp. 10 ( 1976) 239.
FIEiAL LETTERS ELSEVIER
Chemical Physics Letters 233 ( 1995) 399-404
A close-coupling study of collision-induced dissociation inHe+H, K. Nobusada a, K. Sakimoto a, K. Onda b aInstitute of Space and Astronautical Science, Yoshinodai, Sagamihara 229, Japan ’ Faculty of Industrial Science and Technology, Science University of Tokyo. Oshamanbe, Hokkaido 049-35, Japan Received 26 September 1994; in final form 2 December 1994
Abstract The previous quantum mechanical close-coupling study of the dissociative collisions in He + Hz (T-shape configuration) by the present authors [Chem. Phys. Letters 216 (1993) 6131 has been extended to a wide range of the total energy 4.8-7.5 eV. The dissociation process shows a clear vibrational enhancement. The dissociation probability has an undulation as a function of the total energy, and by using classical S-matrix theory it is shown that this feature is caused by interference of the contributions of multiple trajectories.
1. Introduction
In atom-diatom collisions at energies above the dissociation threshold, we have a double continuum of scattering and dissociative states. Since it is not easy to solve the quantum mechanical problem of double continuum states, not so many elaborate quantum mechanical calculations have been made for dissociative collisions [ l-121. When a collision system has no rearrangement channel, it is reasonable to regard the dissociation of a molecule merely as a vibrational excitation to continuum states. In this case, the collision process can be described in the framework of the standard close-coupling (CC) method, in which the total wavefunction is expanded in terms of the vibrational eigenfunctions of the target molecule. Diestler and co-workers have first applied the CC method to the dissociation process [ 31. Following their work, the present authors have applied the CC method to the dissociative He + H, collisions on a realistic potential energy surface [ 111 (hereafter 0009-2614/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01462-O
referred to as paper I), and have found that the CC method can accurately describe the dissociation process. In paper I, we have considered He + H2 collisions only at some limited total energies just above the dissociation threshold ( = 4.75 eV) . In this Letter, we have extended the calculation to a wide range of the total energy (4.8-7.5 eV) and have studied the energy dependence of the dissociation probabilities. As in paper I, we have assumed that the He atom approaches along the line bisecting the H2 internuclear distance and all the particles remain in a plane during collisions: i.e. the T-shape configuration. In order to obtain a deeper understanding of the collision dynamics, we have also made classical and semiclassical calculations.
2. Theory We employ the notation as described in paper I. Let distance of HZ, and R be the distance between He and the center-of-mass of H,. The r be the internuclear
400
K. Nobusada et al. /Chemical
Schrijdinger equation for the He + H, system is given by ti2 a2 -----2~ aR2 =Ep(R,
,“,: ,“:l + V(R, r) + V,(r) r) ,
(1)
(-j$+k$1
xn,,n(R) (2)
where kfg is the channel wave number defined by ki, = (2p/h2)(Ea,) with E,, being the vibrational energy of H2 for 0 G n’ Q 14 and being the energy eigenvalue of discretized continuum states for n’ > 15, and (r$,,r 1VI &” ) means the integral over r. The scattering s-matrix elements S,.,n are obtained by imposing the usual boundary condition on x~,,,( R) in the asymptotic region of R + ~0.The probability for the transition from the initial vibrational state n to the final state n ’ is P,,, = ]S, P,n]2.The dissociation probability for the initial state n is given by =
c PI’>15
p,*,,
We define the kinetic energy distribution of the H atom produced by the dissociation of the H, molecule in the initial vibrational state n as
WR, r)
where m and Al.are the reduced masses of H, and of the collision system, respectively; V(R, r) + V,,(r) is the potential energy surface of the T-shape He + H2 system, V,(r) being the potential energy of H,, and E is the total energy measured from the bottom of the H, well. The interaction potential V(R, r) becomes zero as R+m. We have used the analytic fit of the ab initio potential given by Varandas and Brandb [ 131 for V(R, r) + V,(r). The contour map of the potential energy surface is shown in paper I. As in paper I, we discretize the dissociative continuum by assuming that the vibrational eigenfunctions of H2 always vanish outside a box with size r,,. We must take the box size r,, sufficiently large so that the calculated transition probabilities become independent of r_. We set r,, = 15.5 bohr, which is sufficiently large to obtain the accurate transition probabilities reported in this paper. Using the eigenfunctions c#+,( r) of the H2 bound states (0 < n Q 14) and of the discretized H2 dissociative states (n > 15)) we expand the total wav&,(r)X,.,,(R).Then,we efunctionas!P”(R,r)=C,, obtain a set of coupled equations in the form
pdiss ”
Physics Letters 233 (1995) 399-W
.
(3)
(n’> 15) .
(4)
The kinetic energy Ek of the dissociative fragment H is measured in the center-of-mass system of H2, i.e. Ek = 4( E,,,- D), where D is the dissociation energy of the H, molecule ( =4.75 eV). The density of states den,./dn’ has been calculated from the discretized energy levels by using a finite difference formula. Integration of the distribution over the kinetic energy Ek gives the dissociation probability.
3. Results Fig. 1 shows the dissociation probabilities for n = 1, 3,5,7 and 9 in the energy range E = 4.8-7.5 eV. This figure clearly indicates a vibrational enhancement: that is, at a fixed total energy the dissociation probability becomes larger with increasing initial vibrational quantum number. As is often used in other papers, we introduce a loose definition of the dynamical threshold, above which the dissociation probability becomes noticeable. From Fig. 1, we see that the dynamical threshold becomes higher for lower n. This trend in the dissociation probability curve is characteristic of a system that has no rearrangement channel [ 10,14-201. The present work shows that as n increases, the dynamical threshold gets closer to the dissociation threshold. A similar conclusion has also been obtained in classical trajectory calculations [ 18,201. It is interesting that the probability curves for the higher vibrational states shown in Fig. 1 have ‘shoulders’ (or undulation). For a collision system that has no rearrangement channel, the shoulder structure has been also found in semiclassical [ 14,151 and quantum mechanical [ lo] calculations. However, no physical explanation has been given for the presence of the shoulder structure. To find out the physical reason, we have investigated the kinetic energy distribution of the H atom produced by the dissociation.
K. Nobusada et al. /Chemical Physics Letters 233 (1995) 399-404
401
(a) E = 5.62 eV I2
0.01 (b) E = 6.43 eV
i
(cl E = 6.92 eV
“.05 s.0
5.5
6.5
6.0
E
20-
1.5 -
IO-
7.0
:
(ev)
Fig. 1.Dissociation probabilities for n = 1,3,5,7 and 9 as a function of the total energy.
Fig. 2 shows the kinetic energy distribution (4)) as an example, for n =7 at three total energies: (a) E = 5.62 eV (just below the first shoulder position in Fig. 1) ; (b) 6.43 eV (just below the second one) and (c) 6.92 eV (just above the second one). At E = 5.62 eV, only a single peak is present. When E = 6.43 eV, this peak moves towards higher Ek,and another peak appears at lower Ek When E = 6.92eV, these two peaks go to the high energy region and a third peak grows at Ek= 0 eV. Here, we call such peaks peak 1, peak 2 and peak 3 as indicated in Fig. 2. It should be noted that the kinetic energy distribution shows another oscillatory pattern at each time after an additional shoulder appears in the Pfssversus E curve. Therefore, we can expect that the oscillatory pattern in the kinetic energy distribution is closely related to the shoulder structure in the dissociation probability. To make this clear, we show the energy dependence of the area of peak 1, peak 2 and peak 3 in Fig. 3. The sum of the areas over all the peaks is the dissociation probability for n = 7. The area of peak 1 increases with
02
0.4
06
08
1.0
E, WI Fig. 2. Kinetic energy distributions of the dissociative fragment H, dP/dE, versus Ek, for n = 7 at (a) E = 5.62 eV; (b) E = 6.43 eV and (c) E=6.92 eV.
energy up to E = 5.8 eV, and has a constant nearly value of about 0.4 beyond E = 5.8 eV. This makes the first shoulder at 5.7 d E< 6.0 eV, as seen in Fig. 1. As the total energy increases further from E = 6.0 eV, the area of peak 2 becomes appreciable. This area increases and becomes nearly constant ( = 0.1) above E = 6.6 eV. This corresponds to the second shoulder at 6.4 GE < 6.7 eV. Thus we can conclude that the shoulder structure results from the oscillatory pattern in the kinetic energy distribution.
4. Classical and semiclassical calculations To understand the reason for the oscillatory pattern seen in the kinetic energy distribution we have also carried out a calculation using the classical S-matrix theory developed by Miller [ 21,221 and Marcus [ 231.
402
K. Nobusada et al. /Chemical Physics Letters 233 (1995) 399-404
spondingly, two classical trajectories contribute to the 12,-+ n2 transition. Then, the semiclassical transition probability for ni + n2 is calculated by [ 21,221
Peak 1
PE”, = /c
[2di
(!$J”’
0.2
where @( n2, ni) is an action integral along the trajectory and the sum is taken over the two possible values of ql. Owing to the presence of the phase factor @( n2, ni ) , an interference effect can be taken into account in classical S-matrix theory. If we neglect the interference effect, we have the classical transition probability in the form
CG $ .%
0.2
a"
Peak 2
0.1
(6)
0.a
i.0
/ 5.5
I
I
6.0
6.5
E
/
,
7.0
E = 5.62 eV
(ev)
Fig.3.Areasofpeakl(O),peak2(A)andpeak3(W)forn=7 as a function of the total energy. For the definition of peaks 1,2 and 3, see Fig. 2.
(Another kind of semiclassical approach may be applicable to treat this problem [ 241.) Rusinek and Roberts [ 14,151 first applied classical S-matrix theory to the study of dissociation processes. Here, we follow their method. For the description of the dissociative continuum, we have employed the box-discretization method [ 14,151, which is just the same as that used in the present quantum mechanical calculation. We introduce the quantities nl and q, which characterize the classical vibrational motion of H2 before collision: n, is the classical correspondence of the vibrational quantum number and q, is the initial value of the angle variable conjugate to the action. Let n2 be the corresponding quantity after collisions. We must seek classical trajectories which have non-negative integers for ni and n2. This can be achieved by plotting n2 as a function of q, for a given non-negative integer n,. Fig. 4 shows an example for n, = 3, 5 and 7 at E = 5.62 eV. If n2 takes a non-negative integer below its maximum, we always have two roots of ql, Corre-
0.0
0.2
0.4
0.6
0
91/2x Fig. 4. Classical trajectory calculations of the final vibrational quantum number for n, =3, 5 and 7 at E=5.62 eV as a function of the initial angle 4, divided by 2a.
K. Nobusada et al. /Chemical
(a) E = 5.6’2 eV
0.0 20
(b) E = 6.43 eV
O?
403
the oscillatory pattern in the kinetic energy distribution is caused as a result of the interference of the two classical trajectories. This interference effect is just the same as the one pointed out by Miller for the boundbound vibrational transition [ 221. The present work shows that the interference effect is also significant in the bound-continuum transition.
I
*
I
0.1
Physics Letters 233 (1995) 399-404
0.3
0.4
0.5
: : :
5. Conclusion An oscillation is seen in the kinetic energy distribution of the dissociative fragment H. This oscillatory pattern makes a shoulder structure in the probability curve for the dissociation. Using classical S-matrix theory, we have shown that the oscillation in the kinetic energy distribution is due to the interference of two classical trajectories. Therefore, we can conclude that the shoulder structure is an interference effect. The shoulder structure has been seen also in the dissociation of the collinear configuration [ 10,141. Probably, this is an interference effect as well. We can expect that the interference effect in the dissociation is present for all the configurations. It is interesting to see whether the interference effect is observed in 3D calculations.
Fig. 5. Kinetic energy distributions of the dissociative fragment H, dP /dE, versus Ek. for n = 7 calculated by quantum mechanical (O), semiclassical (0) and classical (A) methods at (a) E= 5.62 eV; (b) E=6.43 eV and (c) E=6.92 eV.
The dissociation probability and the kinetic energy distribution are calculated in the same way as in Eqs. (3) and (4)) respectively. Here, we have considered only classically allowed transitions. In the actual calculation of Eq. (5), we have used the uniform approximation
Acknowledgement The authors would like to thank Professor Y. Itikawa for his reading the original manuscript carefully. This work is partially supported by a Grant-in-Aid for Scientific Research on Priority Area “Theory of Chemical Reactions” from the Ministry of Education, Science and Culture of Japan.
[221. Fig. 5 shows the kinetic energy distribution of the dissociative fragment H for n = 7 calculated by semiclassical (open circles) and classical (open triangles) methods. The closed circles are the results of the quantum mechanical calculation shown in Fig. 2. The semiclassical distribution agrees well with the quantum mechanical one except for the high kinetic energy region Ek, in which the transition to these states is classically forbidden. However, the classical method, neglecting the interference effect, fails to reproduce the oscillatory pattern. From these facts, it is evident that
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