Centro di Studio per la Chimica dei Plasmi del CNR,
classical, including rotations and vibrations, with some pseudo-quantization rules. The method has generally a sufficient reliability, especially when averaged quantities are required, in which quantal behaviour is to some extent less important (in this sense is surely much safer to obtain cross sections rotationally averaged, as discussed in the preceding paragraph). An important limitation is represented by treating vibration as a classical motion: in other semiclassical methods this is overcome by introducing a Schroedinger equation only for vibration, with remaining degrees of freedom treated classically [13].
All the rovibrational states (348) of H2 supported by the PES have been calculated using the WKB method [14]. Really only about one half of these states are bound: those higher in energy with respect to dissociation limit are "quasibound", that is classically bound, because of the presence of the rotational barrier, but with non-zero quantal escaping probabilities.
These states are of interest in recombination kinetic schemes in which orbiting resonance theory is applied [15]. We modified the standard procedure of pseudoquantization of initial states, by allowing classical rovibrational actions not to be discretized, but distributed in the neighbourhood of each initial state of interest, that is:
VÎ [Vo-h/2,Vo+h/2], LÎ [Jo-h/4p,Jo+h/4p], assuming the pseudoquantization of initial states given by:
Vo=(v+1/2)h, Jo=(j+1/2)h/2p.,
in which v and j are quantum numbers, Vo and Jo are pseudoquantized classical actions.
Really our code allowes to choose the distribution around the pseudoquantized states (i.e., a trajectory is assigned to a certain state as a function of its "distance" from the nearer pseudoquantized). We have used for this work a simple "square window" function centered in the state of interest. Analogously, also the products are analyzed with the same weight function. A continuos distribution is assigned to translational energy, which ranges from 10–3 eV to 6 eV. There are limitations in reliability of QCT results for both the extrema of this energy interval, because De Broglie wavelength is not negligible at very low translational energies, especially for hydrogen, in comparison with typical interaction distances, while for high energies interactions with other potential energy surfaces should be considered, for example with surface hopping method [16]. Cross section dependence on translational energy is performed by using a large number of bins (150) and averaging with a simple smoothing technique, which calculates the mean value over three or five bins around that one of interest.
Integration of motion is performed by a variable time step Runge-Kutta fourth order method, with total energy check (6*10–4 eV) and 1 in ten back-integration check. Stratified sampling was applied to impact parameter. Not less than 50 000 trajectories were calculated for each initial rovibrational state.
3. Results
Some cross section results for vibrational deexcitation:
H+H2(v,Trot) -> H+H2(v',all j')
are displayed in fig.1, for rotational temperatures of 300K, 5000K and 20 000K. These data include both reactive and non-reactive collision processes, because there is no interest in separating the two cases in kinetic studies for which these data are calculated. At 300K results show relevant statistical fluctuations (about 0.5Å2 root mean square error), due to the fact that
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Figure1. Some vibrational deexcitation cross sections as a function of translational energy, averaged on the rotational temperatures indicated in the figure. (a) initial v=5, final v'=4; (b) v=9, v' = 8; (c) v=12, v'=11; (d) v=14,v'=13.
in this case the thermal average is made over few rotational states relatively noisy. In fact, fluctuations are much smaller for higher temperatures. Actually this does not seem a serious problem: rate coefficients obtained from these results are in good agreement with other data of both experimental and computational origin, as will be shown.
The general trend shown in fig.1 is a rapid increasing of cross section for very low
approximately flat trend (in the limit of statistical fluctuations). For increasing initial vibrational quantum number v the peak increases in value and decreases in energy. Peak position is comparable with the energy interval between adjacent vibrational states. Rotational temperature is more important (roughly 30% of maximum difference in cross section value between 300K and 5000K within 1–2 eV of translational energy) for low initial v than for high one, simply because there is a wide ladder of rotational states in the first case, and consequently a large contribution to the cross section for sufficiently high rotational temperature, being true the opposite for the last vibrational levels. The general trend of dissociation cross sections:
H+H2(v,Trot) -> 3H
are quite different. In that case (fig.2) there is a clear threshold to the process (of course, in this classical computational scheme no tunneling is allowed). Thresholds show a strong dependency from rotational temperature, decreasing with an increasing of this last. For higher energy values there is an increasing section, and the maximum value, reached at the end of the energy interval, is strongly dependent on rotational temperature, increasing with this last.
Dissociation cross section maximum is also much higher for high initial v than for low one.
Comparison with other results in literature is possible for a small number of vibrational deexcitation [17–22] and dissociation [23] cross sections, while it is easier to compare rate coefficients obtained from those data. In fig.3 there are state to state rate coefficients from [6]
(directly calculated, not via cross sections, see [6]) and those calculated by us using cross sections and the following relation:
kv ,v'(Ttr,Trot)Qr1(v) jgj (2 j1) exp(Ev, j / kTrot) (k3Ttr3 / 8)1/ 2 dE E exp(E / kTtr) v, j, v'(E )
with
Qr1(v) jgj (2 j1) exp(Ev, j/ kTrot)
sv,j,v' is the state to state cross section summed over final j' values. The comparison generally good at 2000K and 4000K of ro-translational temperature, while our results are one order of magnitude lower at 500K (it might be an effect of our distributed actions in the reactants, probably more important for low j values). Concerning dissociation cross sections, a good agreement is found with data in [23]. Moreover the global experimental rate quoted in [24] is in good agreement with the corresponding theoretical kinetic quantity based on the present cross sections [25]. As a consequence, fluctuations in cross section results should not be considered very relevant, especially when cross sections are used in kinetic calculations, because any kinetic study is generally performed on a relatively large translational energy interval, so results are in any case averaged on this interval.
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Figure 2 (a–h). Dissociation cross sections as a function of translational energy from a given vibrational state, averaged on a rotational distribution at the temperatures indicated in the figures. (a) v=0, (b) v=1, (c) v=2, (d) v=3, (e) v=4, (f) v=5, (g) v=6, (h) v=7.
Figure 2 (i–o). (i) v=8, (j) v=9, (k) v=10, (l) v=11, (m) v=12, (n) v=13, (o) v=14.
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Figure 3. Comparison of vibrational deexcitation rate coefficient, averaged on a rotational distribution, calculated in this work and from ref.[6]. (a,d) Trot=500K, (b,e) Trot=2000K, (c,f) Trot=4000K, (a,b,c) v=5, (d,e,f) v=9.
10-25 10-23 10-21 10-19 10-17 10-15 10-13
1 10 -4 2 10 -4 3 10 -4 4 10 -4 5 10 -4 6 10 -4 7 10 -4 8 10 -4
C.D.Scott this work
Dissociationrate(cm3 s-1 )
1/T (K -1)
Figure 4. Comparison of global dissociation rate coefficient calculated in this work and obtained experimentally in ref. [24].
4. Conclusions
In this work the complete set of vibrational deexcitation and dissociation cross sections for reactive and non-reactive atom-molecule collisions in hydrogen has been calculated, from any initial rotational state. Data have been then averaged on rotation by using a Boltzmann rotational distribution. The results, which are of large applicability in kinetic codes in which translational-vibrational non-equilibrium is studied, have been compared with different data of experimental and computational origin, with a good agreement on averaged quantities (rate coefficients), confirming a good reliability of the dynamical method (QCT with continuos distributions of rovibrational actions) used for this kind of massive computations. However there are some problems if a larger accuracy is required. First of all, using the same computational scheme, a larger number of trajectories could be useful for low value cross sections. At least low lying vibrational states should be treated quantally with a semiclassical method, translation for very low energy values is not reliable when treated classically, and for total energies larger than about 3eV interaction with other potential energy surfaces should be considered. However, the present results can be considered among the most complete sets of cross sections in the literature.
ACKNOWLEDGEMENTS
This work was supported by MURST (9903102919-004) and by ASI (progetto termodinamica, trasporto e cinetica di plasmi H2 per propulsione spaziale).
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