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A semiclassical study of dissociation dynamics in He + H2 collisions
Chemical Physics 236 Ž1998. 123–132
A semiclassical study of dissociation dynamics in He q H 2 collisions Kazuhiro Sakimoto
1
Institute of Space and Astronautical Science, Yoshinodai, Sagamihara 229, Japan Received 4 May 1998
Abstract A semiclassical method, which treats the vibrational motion quantum mechanically and the relative motion classically, is applied to study collision induced dissociation in He q H 2 . The semiclassical method provides a reasonable agreement with previous full quantum mechanical results wK. Nobusada, K. Sakimoto, Chem. Phys. 197 Ž1995. 147x, and is very useful to gain a deep understanding of the dissociation dynamics. The method also enables us to discuss some quantum mechanical effects found in the full quantum mechanical study. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Collision induced dissociation ŽCID. of molecules A q BC ™ A q B q C, is an important process in the hyperthermal energy Ž) a few eV. region. Since a huge number of rotational and vibrational channels participate in such the collisions, a rigorous three dimensional Ž3D. treatment of the CID process is significantly difficult in quantum mechanics. Very recently, some progress has been made in the quantum mechanical study of CID although an infinite-order-sudden ŽIOS. or vibrational-sudden approximation has been introduced w1–6x. The quantum mechanical investigation should be further promoted for the CID process: Previous CID studies have shown that the quantum mechanical interference effect is present w7x, and the quasiclassical trajectory ŽQCT. method may not be valid at least in nonreactive collisions for low vibrational states because the classically forbidden motion im1
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portantly contributes to the CID process w3x. The quantum mechanical treatment will be especially inevitable when we investigate the reverse process, i.e., three-body recombination ŽTBR., which is prominent at very low temperatures. The present paper, as one possibility of dealing with quantum mechanical effects in CID Žor TBR., considers a semiclassical treatment, in which the internal motions are treated quantum mechanically and the relative motion is classically. This type of semiclassical approach has been applied to chemical reaction processes w8–12x. By incorporating further the wavepacket-propagation Žgrid representation. technique, the semiclassical method can be numerically efficient to describe the dissociative motion of molecules w9–11x. The present paper examines to what extent the semiclassical method is applicable to the CID process. For this purpose, the calculations are carried out for some restricted collision configurations in the nonreactive He q H 2 system. Since the full quantum mechanical calculations are available for this system w13x, we can evaluate the accu-
0301-0104r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 2 0 8 - 0
K. Sakimotor Chemical Physics 236 (1998) 123–132
124
racy of the semiclassical method. A discussion is also made for some quantum mechanical effects of the dissociation dynamics.
2. Theory and calculations 2.1. Semiclassical theory We consider He q H 2 collisions with the reduced masses m of the total system and m of the molecule. We introduce Jacobi coordinates R and r, which are the intermolecular distance of He–H 2 and the internuclear distance of H 2 , respectively. We assume that the collisions occur in a restricted configuration, that is, the orientation angle g between the vectors R and r is fixed. Then, the Hamiltonian of the total system is given by Hs
1 2m
PR2 q Ho ,
Ž 1.
where PR is the momentum of the intermolecular motion and "2 E 2 q V Ž R ,r ,g . 2m E r2 with V Ž R,r,g . being the potential energy surface ŽPES. of He q H 2 . In the present calculation, only the electronically ground PES is considered. We treat the vibrationalrdissociative motion Ž r . quantum mechanically and the relative motion Ž R . classically. To do this, the time dependent Schrodin¨ ger equation is employed, i.e., Ho s y
E i"
Et
c Ž r ,t . s Ho c Ž r ,t . ,
Ž 2.
and the time dependence of the classical variable RŽ t . is determined by a common trajectory obeying the energy conservation, Es
1 2m
PR2 q ² c < Ho < c : ,
Ž 3.
where E is the total energy Žhereafter, energies being measured from the bottom of the H 2 potential curve. and ²<: means the integration over r. The use of the above common trajectory has the defect that the detailed balance cannot hold between
the forward and backward transitions. Furthermore, the common trajectory treatment may not be valid for the transition that has large energy transfer, such as dissociation. Some device was suggested to reduce these deficiencies w14x. However, the present study introduces no such efforts for the easiness of numerical calculations, and tests how much the simple common-trajectory assumption holds for the CID process. 2.2. Solution method In order to solve Eq. Ž2. accurately at energies above the dissociation limit D Žs 4.75 eV., we employ a grid representation algorithm w15x instead of the basis-expansion types. This technique is also the semiclassical version of a direct numerical method proposed by the present author and Onda w16x for the full quantum mechanical treatment of CID. We define a equally spaced grid in the r coordinate, r j s jD r , j s 0, PPP , N, where rN s rmax is taken large enough so that the vibrational continuum motion can be described in a sufficient accuracy. We introduce a set of grid-based functions u j Ž r ., which are constructed from orthogonal polynomials with quadrature points r j , and satisfy the Kronecker delta property u jX Ž r j . s d jX , j . In the present calculation, we take w16,17x uj Ž r . s
N
2 Nq1
Ý sin ls1
ž
p lj Nq1
/ žŽ sin
p lr N q 1. D r
/
.
Ž 4. Using this function, we can expand the wavefunction c in the form
c Ž r ,t . s Ý c jX Ž t . u jX Ž r . ,
Ž 5.
X
j
where we have put c jX Ž t . s c Ž r jX ,t .. Inserting this equation into Ž2. and setting r s r j , we have a set of coupled linear differential equations with respect to c j Ž t ., d cj Ž t . dt
N
i" s 2m i y "
Ý X
j s1
d 2 u jX Ž r j . dr2
c jX Ž t .
Vj Ž R Ž t . ,g . c j Ž t . ,
Ž 6.
K. Sakimotor Chemical Physics 236 (1998) 123–132
where Vj Ž R,g . s V Ž R,r j ,g .. The coupling matrix elements d 2 u jX Ž r j .rd r 2 are purely mathematical and are independent of collision systems. In the present method, since the wavefunction c Ž r,t . is directly obtained, we can easily monitor the time evolution of the local distribution
r Ž r ,t . s < c Ž r ,t . < 2 .
Ž R Ž t . ,r Ž t . .
ž
t
X
a nonnegligible value. The choice of rmax s 15 bohr is reasonable because the amplitude of wavefunction c Ž r,t . is always negligible at r ) rmax in the present calculations. However, a larger value of rmax may be required when we consider very high vibrational states for g s 90 degree w19x.
Ž 7.
Visualization of this quantity provides us with a deep insight of the collision dynamics. Furthermore, as seen later, the average trajectory defined by
s my1
125
X
H P Ž t . d t ,² c Ž t . < r < c Ž t . : / , R
3. Results and discussion Since the dissociation probability is defined by Eq. Ž11., first we check the accuracy of the semiclassical calculation for the vibrational transition. Fig. 1
Ž 8.
is also useful to characterize the collisions. 2.3. Transition probabilities We take the initial condition as
c Ž r ,t s 0 . s x Õ Ž r . ,
Ž 9.
where x Õ Ž r . is the H2 vibrational bound wavefunction normalized to unity. After the collisions, projecting c Ž r,t s `. onto the vibrational bound wavefunction, we can obtain the probability for the vibrational transition Õ ™ ÕX in the form P Ž Õ ™ ÕX . s <² x ÕX < c Ž t s ` . :< 2 .
Ž 10 .
Since the norm ² c Ž t .< c Ž t .: Ži.e., the total probability. is always conservative and is equal to unity, the dissociation probability for Õ may be defined by Õmax
P
diss
Ž Õ. s1y
Ý P Ž Õ ™ ÕX . ,
Ž 11 .
X
Õ s0
where Õmax Žs 14. is the highest vibrational bound quantum number. 2.4. Numerical calculations We employ the PES obtained by the ab initio calculation of Varandas and Brandao ˜ w18x. We have carried out the calculations at E s 5–10 eV for three configuration angles of g s 0, 45, and 90 degree. The transition probabilities Ž10. and Ž11. have been evaluated at R s 8 bohr. We have set rmax s 15 bohr and N s 200, and have solved Eq. Ž6. in a fourthorder Runge-Kutta algorithm only for c j Ž t . that has
X
Fig. 1. Õ s 0 ™ Õ transition probabilities for configuration angles g s 0, 45, 90 degree at Es6 and 10 eV calculated by the semiclassical method. The quantum mechanical results w13x are also shown.
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K. Sakimotor Chemical Physics 236 (1998) 123–132
shows the Õ s 0 ™ ÕX transition probabilities for three configuration angles, g s 0, 45, and 90 degree, obtained by the present semiclassical and the quantum mechanical w13x methods. We can obtain a good agreement for the two methods both at low Ž E s 6 eV. and high Ž E s 10 eV. energies. This fact promises the reliability of the present calculation at least for the ‘‘total’’ dissociation probabilities. Nevertheless, it does not necessarily mean that the assumption of the common trajectory is always valid. A further examination will be needed, for example, when we study the energy distribution of dissociative fragments. In quantum mechanical studies, Nobusada and the present author w2,3,13x have found that the CID process in He q H 2 occurs only in a narrow range of configuration angle Žg ; 90 degree. when the vibrational states are low. This fact can be convinced also in Fig. 1, where the degree of vibrational excitation is very low for the lower angles g s 0 and 45 degree. Therefore, we can expect that the dissociation dynamics is well representative by viewing the collision features in the configuration with g s 90 degree. In the following, a lot more space is spent for the case of g s 90 degree. 3.1. Collisions for g s 90 degree Fig. 2 shows the dissociation probabilities for the initial vibrational states Õ s 0,1, and 5 at energies E s 5–10 eV in the collisions with the configuration angle g s 90 degree. Comparison is made for the present semiclassical and the quantum mechanical w13x calculations. An overall agreement is good. The semiclassical method can reproduce well the undulation structure of the quantum mechanical probabilities seen for Õ / 0. However, the semiclassical method gives an effective dissociation threshold Žhereafter called dynamical threshold. lower than the quantum mechanical one. We have also compared the QCT results w3x for Õ s 0 in Fig. 2. Nobusada and the present author w3x have pointed out that the contribution of the classically forbidden motion is important in the CID process at low energies, and consequently the QCT method provides a too high dynamical threshold as shown in Fig. 2. To see this quantum mechanical picture more deeply, we have drawn the time evolu-
Fig. 2. Dissociation probabilities as a function of E for Õ s 0,1,5. The collision configuration is g s90 degree. The quantum mechanical w13x and QCT w3x results are also shown.
tion of the local distribution Ž7. for Õ s 0 in Fig. 3. At E s 6 eV Žlower than the dynamical threshold., we can see that the wavepacket expands by the collision impact. This actually leads to the vibrational excitation, but not at all to the dissociation yet. On the other hand, at E s 10 eV Žmuch higher than the dynamical threshold., the most part of the wavepacket is scattered into the dissociative continuum region, and the dissociation is a dominant channel. When E s 8 eV, which is nearly equal to the classical dynamical threshold, the tail Ži.e., not the main part. of the wavepacket can contribute to the dissociation. Hence, the semiclassical method gives a finite dissociation probability. The tail component of the wavepacket contains a classically forbidden nature. Therefore, the dissociation dynamics caused by the wavepacket tail is just quantum mechanical, and can never be explained in terms of classical mechanics as found in the previous study w3x. The semiclassical method takes account of this quantum mechanical effect qualitatively, but fails to give an accurate value of the dynamical threshold. It is natural that the semiclassical theory becomes less valid in the threshold region.
K. Sakimotor Chemical Physics 236 (1998) 123–132
127
Fig. 4. Average trajectories Ž RŽ t .,r Ž t .. on the contour map of the PES for g s90 degree and Õ s 0 at Es 4,5,6,7,8,9,10 eV.
tory shows the reflection to larger H 2 distances Ž r ., i.e, more dissociative. This collision picture is well characterized by the structure of the PES near the turning point of the relative Ž R . motion. Namely, in Fig. 3. Time evolution Žcontour map in the t y r plot. of the local distribution r Ž r,t . ŽEq. Ž7. in text. in the collision configuration with g s90 degree for Õ s 0 at Es6,8,10 eV. The time is measured in the atomic unit t o s 2.42=10y1 7 s. R tp is the turning point of the relative Ž R . motion.
Fig. 2 shows that the dynamical threshold for Õ s 0 is much higher than the dissociation limit D Žs 4.75 eV.. The same threshold behavior is found also in the 3D calculations when the system has no rearrangement channels w1–3,6,20–22x. In the case of He q H 2 , also to understand the 3D threshold behavior, the examination of only the g s 90 degree collisions would be sufficient because the contribution to the dissociation cross sections except from g ; 90 degree is negligible w2,3x. As seen below, the topological structure of the PES plays a decisive role not only in the threshold behavior but also in most of the dissociation dynamics. Fig. 4 plots the average trajectories Ž8. on the PES contour map for several energies. As the energy increases, the average trajec-
Fig. 5. Average trajectories Ž RŽ t .,r Ž t .. on the contour map of the PES for g s90 degree at Es 7 eV. The vibrational states are Õ s 0,2,4,6.
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K. Sakimotor Chemical Physics 236 (1998) 123–132
the pattern of reflection toward larger r near the turning point. ŽThis is in contrast to the cases of other g ’s. See later.. The simple reflection pattern confirms the previous finding w2,4,6x that the collisions for g s 90 degree proceed always through the stretch motion of the molecule Žstretch mechanism.. The expansion of the wavepacket is seen in Fig. 3 Žand in Fig. 6., and evidently shows that the dissociation occurs through the stretch mechanism. The time evolution of the local distribution is displayed at several energies for Õ s 5 in Fig. 6. The wavepacket has six antinodes, and this structure is preserved until t s 700 ; 800t o Ž t o s 2.42 = 10y1 7 s, the atomic unit of time.. At t ) 800t o , the wavepacket is largely deformed, and accordingly various transition processes can take place. Of particular interest is that only the outer part of the wavepacket contributes to the dissociation, and the number of the antinode leading to the dissociation increases successively with energy. This implies that the amount of increase in the dissociation probability with energy is large Žor small. when the antinode Žor node. part of the wavepacket begins to participate in the dissociation. Thus, as seen in Fig. 2, the undula-
Fig. 6. Time evolution of the local distribution r Ž r,t . in the collision configuration with g s90 degree for Õ s 5 at Es 7,8,9,10 eV. The time is measured in the atomic unit t o s 2.42= 10y1 7 s.
the two dimensional plot of the equipotential line, the line has more negative inclination near the turning point with increasing energy. ŽIt should be noticed that the left Ž R . axis is drawn to decrease.. Evidently, the inclination of the equipotential line in the turning point region is important to determine the reflection angle. Fig. 5 shows the average trajectories for various vibrational states at E s 7 eV. The equipotential line near the turning point is inclined rather positively for high Õ. However, for higher Õ, the turning point locates at larger r, and there the attractive force fastening the H–H is so weak that the dissociation can take place easily. The common picture in the case of g s 90 degree is that irrespective of energies and vibrational states, the average trajectory shows
Fig. 7. Õ dependence of the dissociation probabilities for g s90 degree at Es 7 eV. The quantum mechanical results w13x are also shown.
K. Sakimotor Chemical Physics 236 (1998) 123–132
tion structure is produced in the dissociation probability curves for high Õ. Using the semiclassical S matrix theory w23x, Nobusada et al. w7x have shown that the interference of two classical trajectories makes this undulation. The present study can explain the origin of the undulation in a different, i.e., purely quantum mechanical manner. It is evident that the number of the undulation is always less than the number of the antinodes of the vibrational wavefunction, i.e., Õ q 1. The present consideration further suggests that the undulation can be also observed when the dissociation probabilities are plotted against Õ. This is actually found in Fig. 7, where the dissociation probabilities are shown for Õ s 0–8 at E s 7 eV. The present semiclassical result has a clear undulation structure. In the quantum mechanical calculation w13x, however the undulation is not so prominent as seen for the energy dependence Žcf., Fig. 2.. 3.2. Collisions for g s 45 degree In Fig. 8, the dissociation probabilities are shown for very high vibrational states Õ s 10 and 12. In contrast to the case of g s 90 degree, the probability
Fig. 8. Dissociation probabilities as a function of E for Õ s10 and 12. The collision configuration is g s 45 degree. The quantum mechanical w13x results are also shown.
129
Fig. 9. Average trajectories Ž RŽ t .,r Ž t .. on the contour map of the PES for g s 45 degree. The results are shown for Õ s 0 at Es10 eV and for Õ s12 at Es6,8.2,9.7,10 eV.
curves have a local maximum at E ; 8 and ; 6.5 eV, respectively for Õ s 10 and 12. Again, an overall agreement is good between the semiclassical and quantum mechanical w13x calculations. However, a further small peak is present at E ; 9.5 eV in the semiclassical results. Fig. 9 shows the average trajectories for Õ s 0 and 12. When the vibrational state is low Ž Õ s 0., the average trajectory has a sharp backward reflection. This result means that any energy transfer processes including also the dissociation are always inefficient for Õ s 0 although the energy is high enough. With use of the vibrational sudden approximation, the present author has revealed that the energy transfer is inefficient at middle angles g ; 45 degree in the nonreactive systems when the vibrational states are low w4,6x. Also by considering the difference of the PES in the turning point regions for g s 45 degree ŽFig. 9. and g s 90 degree ŽFig. 4., we can easily understand the inefficiency of the CID process in the case of g s 45 degree. When the vibrational state is high Ž Õ s 12 in Fig. . 9 , the average trajectory has a quite different behavior depending on the collision energy. This is because a ridge structure is present in the PES when
130
K. Sakimotor Chemical Physics 236 (1998) 123–132
proceeds through the stretch mechanism as for g s 90 degree. The classical motion on the ridge line is expected unstable. Probably for this reason, the semiclassical calculation has a small peak at E ; 9.5 eV in the dissociation probabilities for Õ s 12 ŽFig. 8.. The full quantum mechanical dynamics has no such instability, and hence the probability is a simply increasing function in the energy range E s 9–10 eV. In this way, the mechanism of the dissociation is found different at the low Ž E - 8 eV. and high Ž E ) 9 eV. energies, and owing to this fact, the local minimum is produced in the dissociation probability at the boundary energy Ž E ; 8.5 eV.. In Fig. 10, we show the time evolution of the local density for Õ s 12 at E s 6 and 10 eV, and it can be clearly seen that the different mechanisms of the compression and stretch motions contribute to the dissociation, respectively at the low and high energies. 3.3. Collisions for g s 0 degree In the configuration with g s 0 degree, the collision impact always acts as the compression of the molecule. Fig. 11 presents the average trajectories for Õ s 0 and 7 at E s 10 eV. In the incoming Fig. 10. Time evolution of the local distribution r Ž r,t . in the collision configuration with g s 45 degree for Õ s12 at Es6 and 10 eV. The time is measured in the atomic unit t o s 2.42= 10y1 7 s.
g / 90 degree. The collisions at four energies Ž E s 6,8.2,9.7,10 eV. are shown for Õ s 12 in Fig. 9. At the lower energies Ž E s 6 and 8.2 eV., the turning points locate at a mountainside near the potential valley, and the collisions have the reflection to shorten the H 2 distance there. Evidently, the compression mechanism w2,4,6x dominates the dissociation. In this mountainside region, the equipotential line near the turning point is inclined such that the reflection becomes more backward for higher energies. Therefore, the dissociation is more efficient in the collision at E s 6 eV than at E s 8.2 eV, and accordingly we have the local maximum in the probability at E ; 6.5 eV. At the higher energies Ž E s 9.7 and 10 eV., on the other hand, the turning points lie near a potential ridge line, and the reflection leads to enlarge the H 2 distance. The dissociation in this case
Fig. 11. Average trajectories Ž RŽ t .,r Ž t .. on the contour map of the PES for g s 0 degree for Õ s 0 and 7 at Es10 eV. The results are also shown for the artificial ‘‘1 He’’ impact.
K. Sakimotor Chemical Physics 236 (1998) 123–132
phase, the average trajectories show a bend to very small r just before the turning point: i.e., the molecule is largely compressed at the first stage of the collision. The energy transfer after the collision would be large if the other collision mechanisms were negligible. In the mass combination of He–H– H, however the net of energy transfer is drastically suppressed in the collinear collisions w2,13x. This is because multiple collisions are present w2,13x, i.e, the incident He atom makes a close encounter with the H atom twice or more. The effect of the multiple collision is clearly seen in the example of Õ s 7. The average trajectory has almost the same curve with a large bend angle both for the incoming and outgoing phases. It says that at least one close encounter of He and H is present each in the incoming and outgoing phases. Furthermore, the bend with about the right angle at R s 2.2–3.2 bohr indicates the very efficient energy transfer at each encounter. That is, at the first encounter in the incoming phase, the relative collision energy is mostly transferred to the vibrational motion of H 2 ; next, the molecule is compressed and then stretches; and at the second encounter in the outgoing phase, the vibrational energy is again mostly transferred to the relative motion. Thus, in this collision, the net energy transfer be-
Table 1 Vibrational transition and dissociation probabilities from Õ s 7 at Es10 eV for the normal 4 He and the artificial ‘‘1 He’’ impacts in the collision configuration with g s 0 degree X
Õ s 7™ Õ
4
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 diss
0.0 0.0 0.0 0.0 0.0 0.004 0.013 0.962 0.004 0.013 0.003 0.0 0.0 0.0 0.0 0.0
He
1
He
0.0 0.0 0.003 0.051 0.159 0.014 0.099 0.010 0.021 0.062 0.054 0.026 0.007 0.001 0.0 0.493
131
comes only slight, and the average trajectory is almost symmetric around the turning point. It has been argued that the effect of the multiple collision becomes less significant in an equal-mass system w2,13x. In Fig. 11, hence we also plot the average trajectories for an artificial ‘‘1 He’’ q H 2 system in which the He mass is set to the H one. The average trajectory for Õ s 7 is no more symmetric around the turning point. In particular, both r and R increase monotonically after the first compression of the molecule. Therefore, it suggests that no further significant encounter occurs after the first close encounter. Table 1 shows the vibrational transition and dissociation probabilities for Õ s 7 at E s 10 eV. The results are compared between the normal 4 He and artificial 1 He impacts. The 4 He q H 2 collisions are almost elastic. On the other hand, the ‘‘1 He’’ q H 2 collisions are mostly inelastic and even dissociative. The mass combination is an important factor to determine the dissociation dynamics in the collinear collisions as pointed out previously w2,24,25x and probably also in the 3D collisions w4,6x.
4. Summary and future prospects We have applied the semiclassical method to the CID process in the He q H 2 system that has the restricted collision configuration with g s 0, 45, and 90 degree. The semiclassical results agree reasonably well with the quantum mechanical ones. Visualization of the time evolution of the local distribution is useful to understand some quantum mechanical effects found in the CID process. By examining the average trajectory, we can gain a deep insight of what a part and what a topological structure of the PES is important in the CID process. We can expect that the semiclassical method is powerful to discuss the full 3D dissociative collisions. An investigation of the CID dynamics including the molecular rotational motion is particularly interesting. In some cases the rotational motion plays an important role in the CID process. For example, when the vibrational state is very high, since only a small number of the rotational states are allowed as bound states, the rotational excitation can lead to the dissociation. Furthermore, the QCT studies w26,27x suggested that the presence of the quasibound states
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K. Sakimotor Chemical Physics 236 (1998) 123–132
of the molecule BC supported by the centrifugal barrier Žindirect CID process. might enhance the dissociation cross sections compared with the ones neglecting the quasibound states Ždirect CID process.. Whether the direct or the indirect mechanism dominates is a very interesting question especially in considering the TBR w5,28x. Since the indirect CID process includes the resonances, naturally we must treat it in quantum mechanics. Previous quantum mechanical studies w1–6x are unsatisfactory to fully discuss these rotational effects because the IOS approximation has been introduced. Also to investigate the rotational effects in the CID, the semiclassical method will be promising. This study is now in progress, and will be reported elsewhere. Acknowledgements The author would like to thank to Dr Fumihiro Koike for reading the original manuscript. References w1x w2x w3x w4x
B. Pan, J.M. Bowman, J. Chem. Phys. 103 Ž1995. 9661. K. Nobusada, K. Sakimoto, J. Chem. Phys. 106 Ž1997. 9078. K. Nobusada, K. Sakimoto, Chem. Phys. Lett., in press. K. Sakimoto, J. Phys. B: At. Mol. Opt. Phys. 30 Ž1997. 3881. w5x R.T. Pack, R.B. Walker, B.K. Kendrick, Chem. Phys. Lett. 276 Ž1997. 255. w6x K. Sakimoto, Chem. Phys., in press.
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A semiclassical study of dissociation dynamics in He q H 2 collisions Kazuhiro Sakimoto
1
Institute of Space and Astronautical Science, Yoshinodai, Sagamihara 229, Japan Received 4 May 1998
Abstract A semiclassical method, which treats the vibrational motion quantum mechanically and the relative motion classically, is applied to study collision induced dissociation in He q H 2 . The semiclassical method provides a reasonable agreement with previous full quantum mechanical results wK. Nobusada, K. Sakimoto, Chem. Phys. 197 Ž1995. 147x, and is very useful to gain a deep understanding of the dissociation dynamics. The method also enables us to discuss some quantum mechanical effects found in the full quantum mechanical study. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Collision induced dissociation ŽCID. of molecules A q BC ™ A q B q C, is an important process in the hyperthermal energy Ž) a few eV. region. Since a huge number of rotational and vibrational channels participate in such the collisions, a rigorous three dimensional Ž3D. treatment of the CID process is significantly difficult in quantum mechanics. Very recently, some progress has been made in the quantum mechanical study of CID although an infinite-order-sudden ŽIOS. or vibrational-sudden approximation has been introduced w1–6x. The quantum mechanical investigation should be further promoted for the CID process: Previous CID studies have shown that the quantum mechanical interference effect is present w7x, and the quasiclassical trajectory ŽQCT. method may not be valid at least in nonreactive collisions for low vibrational states because the classically forbidden motion im1
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portantly contributes to the CID process w3x. The quantum mechanical treatment will be especially inevitable when we investigate the reverse process, i.e., three-body recombination ŽTBR., which is prominent at very low temperatures. The present paper, as one possibility of dealing with quantum mechanical effects in CID Žor TBR., considers a semiclassical treatment, in which the internal motions are treated quantum mechanically and the relative motion is classically. This type of semiclassical approach has been applied to chemical reaction processes w8–12x. By incorporating further the wavepacket-propagation Žgrid representation. technique, the semiclassical method can be numerically efficient to describe the dissociative motion of molecules w9–11x. The present paper examines to what extent the semiclassical method is applicable to the CID process. For this purpose, the calculations are carried out for some restricted collision configurations in the nonreactive He q H 2 system. Since the full quantum mechanical calculations are available for this system w13x, we can evaluate the accu-
0301-0104r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 2 0 8 - 0
K. Sakimotor Chemical Physics 236 (1998) 123–132
124
racy of the semiclassical method. A discussion is also made for some quantum mechanical effects of the dissociation dynamics.
2. Theory and calculations 2.1. Semiclassical theory We consider He q H 2 collisions with the reduced masses m of the total system and m of the molecule. We introduce Jacobi coordinates R and r, which are the intermolecular distance of He–H 2 and the internuclear distance of H 2 , respectively. We assume that the collisions occur in a restricted configuration, that is, the orientation angle g between the vectors R and r is fixed. Then, the Hamiltonian of the total system is given by Hs
1 2m
PR2 q Ho ,
Ž 1.
where PR is the momentum of the intermolecular motion and "2 E 2 q V Ž R ,r ,g . 2m E r2 with V Ž R,r,g . being the potential energy surface ŽPES. of He q H 2 . In the present calculation, only the electronically ground PES is considered. We treat the vibrationalrdissociative motion Ž r . quantum mechanically and the relative motion Ž R . classically. To do this, the time dependent Schrodin¨ ger equation is employed, i.e., Ho s y
E i"
Et
c Ž r ,t . s Ho c Ž r ,t . ,
Ž 2.
and the time dependence of the classical variable RŽ t . is determined by a common trajectory obeying the energy conservation, Es
1 2m
PR2 q ² c < Ho < c : ,
Ž 3.
where E is the total energy Žhereafter, energies being measured from the bottom of the H 2 potential curve. and ²<: means the integration over r. The use of the above common trajectory has the defect that the detailed balance cannot hold between
the forward and backward transitions. Furthermore, the common trajectory treatment may not be valid for the transition that has large energy transfer, such as dissociation. Some device was suggested to reduce these deficiencies w14x. However, the present study introduces no such efforts for the easiness of numerical calculations, and tests how much the simple common-trajectory assumption holds for the CID process. 2.2. Solution method In order to solve Eq. Ž2. accurately at energies above the dissociation limit D Žs 4.75 eV., we employ a grid representation algorithm w15x instead of the basis-expansion types. This technique is also the semiclassical version of a direct numerical method proposed by the present author and Onda w16x for the full quantum mechanical treatment of CID. We define a equally spaced grid in the r coordinate, r j s jD r , j s 0, PPP , N, where rN s rmax is taken large enough so that the vibrational continuum motion can be described in a sufficient accuracy. We introduce a set of grid-based functions u j Ž r ., which are constructed from orthogonal polynomials with quadrature points r j , and satisfy the Kronecker delta property u jX Ž r j . s d jX , j . In the present calculation, we take w16,17x uj Ž r . s
N
2 Nq1
Ý sin ls1
ž
p lj Nq1
/ žŽ sin
p lr N q 1. D r
/
.
Ž 4. Using this function, we can expand the wavefunction c in the form
c Ž r ,t . s Ý c jX Ž t . u jX Ž r . ,
Ž 5.
X
j
where we have put c jX Ž t . s c Ž r jX ,t .. Inserting this equation into Ž2. and setting r s r j , we have a set of coupled linear differential equations with respect to c j Ž t ., d cj Ž t . dt
N
i" s 2m i y "
Ý X
j s1
d 2 u jX Ž r j . dr2
c jX Ž t .
Vj Ž R Ž t . ,g . c j Ž t . ,
Ž 6.
K. Sakimotor Chemical Physics 236 (1998) 123–132
where Vj Ž R,g . s V Ž R,r j ,g .. The coupling matrix elements d 2 u jX Ž r j .rd r 2 are purely mathematical and are independent of collision systems. In the present method, since the wavefunction c Ž r,t . is directly obtained, we can easily monitor the time evolution of the local distribution
r Ž r ,t . s < c Ž r ,t . < 2 .
Ž R Ž t . ,r Ž t . .
ž
t
X
a nonnegligible value. The choice of rmax s 15 bohr is reasonable because the amplitude of wavefunction c Ž r,t . is always negligible at r ) rmax in the present calculations. However, a larger value of rmax may be required when we consider very high vibrational states for g s 90 degree w19x.
Ž 7.
Visualization of this quantity provides us with a deep insight of the collision dynamics. Furthermore, as seen later, the average trajectory defined by
s my1
125
X
H P Ž t . d t ,² c Ž t . < r < c Ž t . : / , R
3. Results and discussion Since the dissociation probability is defined by Eq. Ž11., first we check the accuracy of the semiclassical calculation for the vibrational transition. Fig. 1
Ž 8.
is also useful to characterize the collisions. 2.3. Transition probabilities We take the initial condition as
c Ž r ,t s 0 . s x Õ Ž r . ,
Ž 9.
where x Õ Ž r . is the H2 vibrational bound wavefunction normalized to unity. After the collisions, projecting c Ž r,t s `. onto the vibrational bound wavefunction, we can obtain the probability for the vibrational transition Õ ™ ÕX in the form P Ž Õ ™ ÕX . s <² x ÕX < c Ž t s ` . :< 2 .
Ž 10 .
Since the norm ² c Ž t .< c Ž t .: Ži.e., the total probability. is always conservative and is equal to unity, the dissociation probability for Õ may be defined by Õmax
P
diss
Ž Õ. s1y
Ý P Ž Õ ™ ÕX . ,
Ž 11 .
X
Õ s0
where Õmax Žs 14. is the highest vibrational bound quantum number. 2.4. Numerical calculations We employ the PES obtained by the ab initio calculation of Varandas and Brandao ˜ w18x. We have carried out the calculations at E s 5–10 eV for three configuration angles of g s 0, 45, and 90 degree. The transition probabilities Ž10. and Ž11. have been evaluated at R s 8 bohr. We have set rmax s 15 bohr and N s 200, and have solved Eq. Ž6. in a fourthorder Runge-Kutta algorithm only for c j Ž t . that has
X
Fig. 1. Õ s 0 ™ Õ transition probabilities for configuration angles g s 0, 45, 90 degree at Es6 and 10 eV calculated by the semiclassical method. The quantum mechanical results w13x are also shown.
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shows the Õ s 0 ™ ÕX transition probabilities for three configuration angles, g s 0, 45, and 90 degree, obtained by the present semiclassical and the quantum mechanical w13x methods. We can obtain a good agreement for the two methods both at low Ž E s 6 eV. and high Ž E s 10 eV. energies. This fact promises the reliability of the present calculation at least for the ‘‘total’’ dissociation probabilities. Nevertheless, it does not necessarily mean that the assumption of the common trajectory is always valid. A further examination will be needed, for example, when we study the energy distribution of dissociative fragments. In quantum mechanical studies, Nobusada and the present author w2,3,13x have found that the CID process in He q H 2 occurs only in a narrow range of configuration angle Žg ; 90 degree. when the vibrational states are low. This fact can be convinced also in Fig. 1, where the degree of vibrational excitation is very low for the lower angles g s 0 and 45 degree. Therefore, we can expect that the dissociation dynamics is well representative by viewing the collision features in the configuration with g s 90 degree. In the following, a lot more space is spent for the case of g s 90 degree. 3.1. Collisions for g s 90 degree Fig. 2 shows the dissociation probabilities for the initial vibrational states Õ s 0,1, and 5 at energies E s 5–10 eV in the collisions with the configuration angle g s 90 degree. Comparison is made for the present semiclassical and the quantum mechanical w13x calculations. An overall agreement is good. The semiclassical method can reproduce well the undulation structure of the quantum mechanical probabilities seen for Õ / 0. However, the semiclassical method gives an effective dissociation threshold Žhereafter called dynamical threshold. lower than the quantum mechanical one. We have also compared the QCT results w3x for Õ s 0 in Fig. 2. Nobusada and the present author w3x have pointed out that the contribution of the classically forbidden motion is important in the CID process at low energies, and consequently the QCT method provides a too high dynamical threshold as shown in Fig. 2. To see this quantum mechanical picture more deeply, we have drawn the time evolu-
Fig. 2. Dissociation probabilities as a function of E for Õ s 0,1,5. The collision configuration is g s90 degree. The quantum mechanical w13x and QCT w3x results are also shown.
tion of the local distribution Ž7. for Õ s 0 in Fig. 3. At E s 6 eV Žlower than the dynamical threshold., we can see that the wavepacket expands by the collision impact. This actually leads to the vibrational excitation, but not at all to the dissociation yet. On the other hand, at E s 10 eV Žmuch higher than the dynamical threshold., the most part of the wavepacket is scattered into the dissociative continuum region, and the dissociation is a dominant channel. When E s 8 eV, which is nearly equal to the classical dynamical threshold, the tail Ži.e., not the main part. of the wavepacket can contribute to the dissociation. Hence, the semiclassical method gives a finite dissociation probability. The tail component of the wavepacket contains a classically forbidden nature. Therefore, the dissociation dynamics caused by the wavepacket tail is just quantum mechanical, and can never be explained in terms of classical mechanics as found in the previous study w3x. The semiclassical method takes account of this quantum mechanical effect qualitatively, but fails to give an accurate value of the dynamical threshold. It is natural that the semiclassical theory becomes less valid in the threshold region.
K. Sakimotor Chemical Physics 236 (1998) 123–132
127
Fig. 4. Average trajectories Ž RŽ t .,r Ž t .. on the contour map of the PES for g s90 degree and Õ s 0 at Es 4,5,6,7,8,9,10 eV.
tory shows the reflection to larger H 2 distances Ž r ., i.e, more dissociative. This collision picture is well characterized by the structure of the PES near the turning point of the relative Ž R . motion. Namely, in Fig. 3. Time evolution Žcontour map in the t y r plot. of the local distribution r Ž r,t . ŽEq. Ž7. in text. in the collision configuration with g s90 degree for Õ s 0 at Es6,8,10 eV. The time is measured in the atomic unit t o s 2.42=10y1 7 s. R tp is the turning point of the relative Ž R . motion.
Fig. 2 shows that the dynamical threshold for Õ s 0 is much higher than the dissociation limit D Žs 4.75 eV.. The same threshold behavior is found also in the 3D calculations when the system has no rearrangement channels w1–3,6,20–22x. In the case of He q H 2 , also to understand the 3D threshold behavior, the examination of only the g s 90 degree collisions would be sufficient because the contribution to the dissociation cross sections except from g ; 90 degree is negligible w2,3x. As seen below, the topological structure of the PES plays a decisive role not only in the threshold behavior but also in most of the dissociation dynamics. Fig. 4 plots the average trajectories Ž8. on the PES contour map for several energies. As the energy increases, the average trajec-
Fig. 5. Average trajectories Ž RŽ t .,r Ž t .. on the contour map of the PES for g s90 degree at Es 7 eV. The vibrational states are Õ s 0,2,4,6.
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K. Sakimotor Chemical Physics 236 (1998) 123–132
the pattern of reflection toward larger r near the turning point. ŽThis is in contrast to the cases of other g ’s. See later.. The simple reflection pattern confirms the previous finding w2,4,6x that the collisions for g s 90 degree proceed always through the stretch motion of the molecule Žstretch mechanism.. The expansion of the wavepacket is seen in Fig. 3 Žand in Fig. 6., and evidently shows that the dissociation occurs through the stretch mechanism. The time evolution of the local distribution is displayed at several energies for Õ s 5 in Fig. 6. The wavepacket has six antinodes, and this structure is preserved until t s 700 ; 800t o Ž t o s 2.42 = 10y1 7 s, the atomic unit of time.. At t ) 800t o , the wavepacket is largely deformed, and accordingly various transition processes can take place. Of particular interest is that only the outer part of the wavepacket contributes to the dissociation, and the number of the antinode leading to the dissociation increases successively with energy. This implies that the amount of increase in the dissociation probability with energy is large Žor small. when the antinode Žor node. part of the wavepacket begins to participate in the dissociation. Thus, as seen in Fig. 2, the undula-
Fig. 6. Time evolution of the local distribution r Ž r,t . in the collision configuration with g s90 degree for Õ s 5 at Es 7,8,9,10 eV. The time is measured in the atomic unit t o s 2.42= 10y1 7 s.
the two dimensional plot of the equipotential line, the line has more negative inclination near the turning point with increasing energy. ŽIt should be noticed that the left Ž R . axis is drawn to decrease.. Evidently, the inclination of the equipotential line in the turning point region is important to determine the reflection angle. Fig. 5 shows the average trajectories for various vibrational states at E s 7 eV. The equipotential line near the turning point is inclined rather positively for high Õ. However, for higher Õ, the turning point locates at larger r, and there the attractive force fastening the H–H is so weak that the dissociation can take place easily. The common picture in the case of g s 90 degree is that irrespective of energies and vibrational states, the average trajectory shows
Fig. 7. Õ dependence of the dissociation probabilities for g s90 degree at Es 7 eV. The quantum mechanical results w13x are also shown.
K. Sakimotor Chemical Physics 236 (1998) 123–132
tion structure is produced in the dissociation probability curves for high Õ. Using the semiclassical S matrix theory w23x, Nobusada et al. w7x have shown that the interference of two classical trajectories makes this undulation. The present study can explain the origin of the undulation in a different, i.e., purely quantum mechanical manner. It is evident that the number of the undulation is always less than the number of the antinodes of the vibrational wavefunction, i.e., Õ q 1. The present consideration further suggests that the undulation can be also observed when the dissociation probabilities are plotted against Õ. This is actually found in Fig. 7, where the dissociation probabilities are shown for Õ s 0–8 at E s 7 eV. The present semiclassical result has a clear undulation structure. In the quantum mechanical calculation w13x, however the undulation is not so prominent as seen for the energy dependence Žcf., Fig. 2.. 3.2. Collisions for g s 45 degree In Fig. 8, the dissociation probabilities are shown for very high vibrational states Õ s 10 and 12. In contrast to the case of g s 90 degree, the probability
Fig. 8. Dissociation probabilities as a function of E for Õ s10 and 12. The collision configuration is g s 45 degree. The quantum mechanical w13x results are also shown.
129
Fig. 9. Average trajectories Ž RŽ t .,r Ž t .. on the contour map of the PES for g s 45 degree. The results are shown for Õ s 0 at Es10 eV and for Õ s12 at Es6,8.2,9.7,10 eV.
curves have a local maximum at E ; 8 and ; 6.5 eV, respectively for Õ s 10 and 12. Again, an overall agreement is good between the semiclassical and quantum mechanical w13x calculations. However, a further small peak is present at E ; 9.5 eV in the semiclassical results. Fig. 9 shows the average trajectories for Õ s 0 and 12. When the vibrational state is low Ž Õ s 0., the average trajectory has a sharp backward reflection. This result means that any energy transfer processes including also the dissociation are always inefficient for Õ s 0 although the energy is high enough. With use of the vibrational sudden approximation, the present author has revealed that the energy transfer is inefficient at middle angles g ; 45 degree in the nonreactive systems when the vibrational states are low w4,6x. Also by considering the difference of the PES in the turning point regions for g s 45 degree ŽFig. 9. and g s 90 degree ŽFig. 4., we can easily understand the inefficiency of the CID process in the case of g s 45 degree. When the vibrational state is high Ž Õ s 12 in Fig. . 9 , the average trajectory has a quite different behavior depending on the collision energy. This is because a ridge structure is present in the PES when
130
K. Sakimotor Chemical Physics 236 (1998) 123–132
proceeds through the stretch mechanism as for g s 90 degree. The classical motion on the ridge line is expected unstable. Probably for this reason, the semiclassical calculation has a small peak at E ; 9.5 eV in the dissociation probabilities for Õ s 12 ŽFig. 8.. The full quantum mechanical dynamics has no such instability, and hence the probability is a simply increasing function in the energy range E s 9–10 eV. In this way, the mechanism of the dissociation is found different at the low Ž E - 8 eV. and high Ž E ) 9 eV. energies, and owing to this fact, the local minimum is produced in the dissociation probability at the boundary energy Ž E ; 8.5 eV.. In Fig. 10, we show the time evolution of the local density for Õ s 12 at E s 6 and 10 eV, and it can be clearly seen that the different mechanisms of the compression and stretch motions contribute to the dissociation, respectively at the low and high energies. 3.3. Collisions for g s 0 degree In the configuration with g s 0 degree, the collision impact always acts as the compression of the molecule. Fig. 11 presents the average trajectories for Õ s 0 and 7 at E s 10 eV. In the incoming Fig. 10. Time evolution of the local distribution r Ž r,t . in the collision configuration with g s 45 degree for Õ s12 at Es6 and 10 eV. The time is measured in the atomic unit t o s 2.42= 10y1 7 s.
g / 90 degree. The collisions at four energies Ž E s 6,8.2,9.7,10 eV. are shown for Õ s 12 in Fig. 9. At the lower energies Ž E s 6 and 8.2 eV., the turning points locate at a mountainside near the potential valley, and the collisions have the reflection to shorten the H 2 distance there. Evidently, the compression mechanism w2,4,6x dominates the dissociation. In this mountainside region, the equipotential line near the turning point is inclined such that the reflection becomes more backward for higher energies. Therefore, the dissociation is more efficient in the collision at E s 6 eV than at E s 8.2 eV, and accordingly we have the local maximum in the probability at E ; 6.5 eV. At the higher energies Ž E s 9.7 and 10 eV., on the other hand, the turning points lie near a potential ridge line, and the reflection leads to enlarge the H 2 distance. The dissociation in this case
Fig. 11. Average trajectories Ž RŽ t .,r Ž t .. on the contour map of the PES for g s 0 degree for Õ s 0 and 7 at Es10 eV. The results are also shown for the artificial ‘‘1 He’’ impact.
K. Sakimotor Chemical Physics 236 (1998) 123–132
phase, the average trajectories show a bend to very small r just before the turning point: i.e., the molecule is largely compressed at the first stage of the collision. The energy transfer after the collision would be large if the other collision mechanisms were negligible. In the mass combination of He–H– H, however the net of energy transfer is drastically suppressed in the collinear collisions w2,13x. This is because multiple collisions are present w2,13x, i.e, the incident He atom makes a close encounter with the H atom twice or more. The effect of the multiple collision is clearly seen in the example of Õ s 7. The average trajectory has almost the same curve with a large bend angle both for the incoming and outgoing phases. It says that at least one close encounter of He and H is present each in the incoming and outgoing phases. Furthermore, the bend with about the right angle at R s 2.2–3.2 bohr indicates the very efficient energy transfer at each encounter. That is, at the first encounter in the incoming phase, the relative collision energy is mostly transferred to the vibrational motion of H 2 ; next, the molecule is compressed and then stretches; and at the second encounter in the outgoing phase, the vibrational energy is again mostly transferred to the relative motion. Thus, in this collision, the net energy transfer be-
Table 1 Vibrational transition and dissociation probabilities from Õ s 7 at Es10 eV for the normal 4 He and the artificial ‘‘1 He’’ impacts in the collision configuration with g s 0 degree X
Õ s 7™ Õ
4
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 diss
0.0 0.0 0.0 0.0 0.0 0.004 0.013 0.962 0.004 0.013 0.003 0.0 0.0 0.0 0.0 0.0
He
1
He
0.0 0.0 0.003 0.051 0.159 0.014 0.099 0.010 0.021 0.062 0.054 0.026 0.007 0.001 0.0 0.493
131
comes only slight, and the average trajectory is almost symmetric around the turning point. It has been argued that the effect of the multiple collision becomes less significant in an equal-mass system w2,13x. In Fig. 11, hence we also plot the average trajectories for an artificial ‘‘1 He’’ q H 2 system in which the He mass is set to the H one. The average trajectory for Õ s 7 is no more symmetric around the turning point. In particular, both r and R increase monotonically after the first compression of the molecule. Therefore, it suggests that no further significant encounter occurs after the first close encounter. Table 1 shows the vibrational transition and dissociation probabilities for Õ s 7 at E s 10 eV. The results are compared between the normal 4 He and artificial 1 He impacts. The 4 He q H 2 collisions are almost elastic. On the other hand, the ‘‘1 He’’ q H 2 collisions are mostly inelastic and even dissociative. The mass combination is an important factor to determine the dissociation dynamics in the collinear collisions as pointed out previously w2,24,25x and probably also in the 3D collisions w4,6x.
4. Summary and future prospects We have applied the semiclassical method to the CID process in the He q H 2 system that has the restricted collision configuration with g s 0, 45, and 90 degree. The semiclassical results agree reasonably well with the quantum mechanical ones. Visualization of the time evolution of the local distribution is useful to understand some quantum mechanical effects found in the CID process. By examining the average trajectory, we can gain a deep insight of what a part and what a topological structure of the PES is important in the CID process. We can expect that the semiclassical method is powerful to discuss the full 3D dissociative collisions. An investigation of the CID dynamics including the molecular rotational motion is particularly interesting. In some cases the rotational motion plays an important role in the CID process. For example, when the vibrational state is very high, since only a small number of the rotational states are allowed as bound states, the rotational excitation can lead to the dissociation. Furthermore, the QCT studies w26,27x suggested that the presence of the quasibound states
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of the molecule BC supported by the centrifugal barrier Žindirect CID process. might enhance the dissociation cross sections compared with the ones neglecting the quasibound states Ždirect CID process.. Whether the direct or the indirect mechanism dominates is a very interesting question especially in considering the TBR w5,28x. Since the indirect CID process includes the resonances, naturally we must treat it in quantum mechanics. Previous quantum mechanical studies w1–6x are unsatisfactory to fully discuss these rotational effects because the IOS approximation has been introduced. Also to investigate the rotational effects in the CID, the semiclassical method will be promising. This study is now in progress, and will be reported elsewhere. Acknowledgements The author would like to thank to Dr Fumihiro Koike for reading the original manuscript. References w1x w2x w3x w4x
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