Influence of Rotational Relaxation on the Effects of
Translational Nonequilibrium of Gas Mixture in the Shock
Wave Front
S.V. Koulikov
Institute of Problems of Chemical Physics, 142432,Chernogolovka, Moscow reg., Russia
Abstract. A new approach for numerical simulation of rotational relaxation of diatomic molecules in the front of a shock
wave was proposed. Diatomic molecules of one sort in different rotational states were considered as different
components, and change in rotational state of molecules as a result of collision was considered as chemical reaction. The
method was checked for N2 and was used for simulation of gas mixture of light gas (He) and two heavy lowconcentration admixtures (O2 and Xe). It was shown in agreement with previous calculations for the case without
consideration of rotational relaxation that in the front the distributions of relative velocities g for pairs of molecules of
admixtures substantially exceeded their equilibrium values behind the wave at high values of g. As early, this effect of
the superequilibrium was more strong for the case of pairs containing molecules of different admixtures (O2 and Xe).
Although results of simulation have shown approximately 10-fold decrease of the effect of superequilibrium the value of
the effect was still high.
1 INTRODUCTION
Studies of nonequilibrium of a gas in the front of a shock wave are important in many cases. (In this paper the
shock wave front is considered to be a nonequilibrium zone separating the states of the translational and rotational
equilibrium gas ahead of and behind the shock wave.) Studies of relative velocity distributions for pairs of
molecules are particularly important for understanding peculiarities of threshold physicochemical processes initiated
by the shock wave. The investigations were performed in all considered bellow cases by the Monte Carlo method of
unstationary statistical simulation with weighting factors [1-3].
Modeling of a planar stationary shock wave was carried out in a one-dimensional coordinate space and a threedimensional velocity space. Collisions of molecules were considered as collisions of hard spheres. A simulation of
shock wave was carried out early for a gas mixture of light gas (He) and two heavy low-concentration admixtures
(O2 and Xe) with numerical density ratio 200:1:1 at Max number M=4 [4-6]. Rotational relaxation of O2 was not
taken into account. It was shown that in the front the distributions of relative velocities g for pairs of molecules
containing admixtures (Gij) substantially exceeded their equilibrium values behind the wave at high values of g. This
means that, in the front of a shock wave, the frequencies of collisions of pairs of these molecules with high g
increases their equilibrium values behind the wave and this may accelerate some threshold physicochemical process
in which these pairs of molecules participate. The effect of superequilibrium is more strong for the case of pairs
containing molecules of different admixtures (O2 and Xe) (G23). It is equal to 106 for g corresponding to threshold
ED of dissociation of O2 at room temperature ahead of the wave.
2 COMPUTATIONAL MODEL
In the present work simulation was carried out by the Monte Carlo method of unstationary statistical simulation
with variable weighting factors taking into account rotational relaxation. Diatomic molecules of one sort in different
rotational states were considered as different components. Change in rotational state of molecules as a result of
collision was considered as chemical reaction (all details of scheme of calculations see in [1,2, 3 (see scheme 2)].
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
The essence of the method can briefly be described as follows.
The simulated medium is replaced by a system of model particles. At initial instant of time, in accordance with
the initial conditions, the particles have given velocities and are distributed over the cells into which the coordinate
space in question is divided. It is assumed that collisions are pair collisions and can take place with a certain
probability only between the particles located in the same cell.
Molecular motion and intermolecular collisions are uncoupled over the small time interval ∆t. At first, all
molecules move freely over ∆t (phase A) and then the collisions are simulated by fixing the molecular positions
(phase B).
The i-th model particle of component l, denoted Al(i), was characterised by mass ml , velocity c l(i)( ul(i), vl(i), wl(i)),
spatial location xl(i) in a selected system of coordinate and weighting factor ηl(i) . This weighting factor expresses the
number of real molecules represented by model particle. Thus, the concentration of real molecules in the cell j of
volume Vj is given by the following expression:
n l( j ) = ∑η (ji ) / V j .
i
Simulation of phase A is very simple. New spatial location of the model particle Al(i)
x l( i )∗ = x l( i ) + ul( i ) ∆t.
The used parameters of collisions are the total cross sections σlmik of elastic collisions between molecules Al(i) and
Am(k), the cross sections σlm,pq(glmik) in a chemical reaction of an inelastic type
Al( i ) + Am( k ) → A(pr ) + Aq( s ) ,
and energy threshold Elm,pq of the above reaction. Some of them depend on glmik =|cl(i)- cm(k)|.The velocities cl(i) / and
cm(k) / of the original molecules after an elastic collision and the velocities cp(r)* and cq(s)* of molecules Αp(r) and Aq(s)
formed as a result of the reaction are calculated according to the laws of conservation of momentum and energy,
taking into account the thermal effect of the reaction.
The following notations are used bellow:
ϑlmik = max{η l( i ) , η m( k ) },
θ lmik = min{η l( i ) , η m( k ) },
ik
ik /
ik
ik
ik /
σ lmik , pq ≡ σ lm , pq ( g lm
), σ lm
= ∑σ lm , pq , σ lm
* = σ lm
+ σ lm
.
ik
Here, σlmik/ is the total cross section of the chemical interaction for particles Al(i) and Am(k).
At the phase B, the evolution of the system was simulated in several (k’) stages step by step. At the each stage,
only the interaction of pairs of particles in the same cell was assumed in simulation of the evolution of the system
during the time of consideration ∆t*= ∆t/k’. Simulation of such a step was carried out using the ballot-box scheme.
For this, all the pairs of particles in a cell were subdivided into general populations in terms of the particle grades for
the formation of pairs. For example, first general population was ensemble of pairs of component 1, second - of
pairs containing one particle of component 1and one particle of component 2 and etceteras. A single pair of particles
Al and Am was randomly selected from the general population (henceforth the indices of the number of particles and
cells will be dropped wherever possible for the sake of simplicity). The evolution of the state of the pair was
simulated by the scheme presented bellow.
Step 1. The interaction of particles Al and Am was simulated with probability
*
Qim = K lmϑlmσ lm
g lm ∆t * / V .
Here Klm is the number of pairs in the general population. If the trial results is negative, the subsequent steps were
omitted.
Step 2. ∆lm=θlmσlm//σlm* was estimated. The weighting factors (weights) of the particles under considerations
were decreased by the value of ∆lm i.e. ηk*=ηk-∆lm (k=l,m).
Step 3. Particles Al and Am with reduced weights collided elastically. The velocity of particle Ak was replaced by
/
ck with probability θlm*/ηk* (k=l, m) and θlm*=min{ ηl*,ηm* }).
Step 4. With the probability σlm,pq/σlm/ use is made of components p and q which are the products of an
appropriate reaction. The numbers of particles r and s of components p and q were randomly selected out of the
totality of the particles of these components in the cell under investigation. (New particles with zero weights were
introduced in the absence of the particles of required components.) The weights of the particles Ap and Aq were
increased by the value of ∆lm i.e ηk*=ηk +∆lm (k=p,q).
Step 5. The velocities of the particles Ap and Aq were replaced by cp* and cq* with probabilities ∆lm/ηp*
and ∆lm/ηq* respectively.
This is so called the improved ballot-box scheme of simulation of the collision phase.
Energy exchange without change in rotational energy of one of colliding molecule can be described by reaction:
Al( i ) + Am( k ) → A(pr ) + Am( k ) .
The cross section for these rotational-translational transitions (RT) were taken from [7] as the power law almost
in the same form, but coefficients Bp were introduced for transitions in several groups of rotational states j. Values of
Bp were different from 1 only for transitions from upper to lower states. A formula for σlm,pm is:
σ lm, pm = R ⋅ B p ' C ( 2 p'+1)( E pm / E lm )1 / 2 2.0 [ p' ( p'+1) − l ' (l '+1)]
−δ
.
(1)
Here δ=0.84, Esq is the energy of relative motion of pair of As and Aq. At start C was equal to 23.3Å2.Then the value
of C was recalculated taking into account that distance along the stream was normalized to the mean free path in the
flow ahead of the wave λ. Index l’ was shifted in comparison with index l so that it was equal to 0 for the lowest
rotational state of diatomic molecule. It should be noted that the total interaction cross section σ*lm of pair of
particles Al and Am was equal to gas-kinetic cross section of their collisions according to the used model of hard
spheres. So, it does not permitted that σlm,pm>σ*lm, otherwise, σlm,pm=σ*lm. It was also assumed that σlm,pm=0 if Elm is
less than threshold of reaction Elm,pm when l’<p’.
An attempt was also made to take into account rotational-rotational transitions (RR) of one type due to collisions
of diatomic molecules :
Al( i ) + Al(−k2) → Al(−r1) + Al(−s1) .
They were chosen because in all these cases the change of rotational energy is equal to 2kχr. Here k is the
Boltzmann constant, χr is characteristic rotational temperature of diatomic molecule. It was assumed that the
reaction occurred if the energy of relative motion of colliding molecules along their center line at the moment of
impact was higher or equal to the threshold of the reaction. In this case the probability Plm,pq of the formation of
particles Ap and Aq due to the collision of particles Al and Am was assumed to be constant; otherwise, Plm,pq=0. The
cross section of reactions in the model under investigation was given by equation [8]:
σ lm, pq
*
2
σ lm
)],
Plm , pq [1 − 2 E lm , pq /( µ lm g lm
µ lm g lm2 / 2 ≥ E lm, pq ,
=
0,
µ lm g lm2 / 2 < E lm, pq ,

were Plm,pq=Sp Sq /Rlm, Sp=2p’+1, Rlm is the number of possible reactions due to the collision of particles Al and Am
taking into account degeneracy of rotational levels. Value of Elm,pq is equal to Rr=0.001kχr for exothermic reaction
and Elm,pq= Rr + 2kχr for endothermic reaction.
The line segment representing the simulation region was subdivided into 160 equally spaced cells ∆x=0.15λ. The
velocity and rotational distribution functions of particles on the boundaries of the simulation region were assumed to
be equilibrium according to the upstream and downstream temperatures. The time of splitting of collision and
displacement stages was ∆t=0,04λ/ν, where v is the most probable thermal velocity of the lightest component
particles in the undisturbed upstream flow. Sampling of parameters of stream took place in time interval 2∆t.
At the start a shock wave was specified in the form of a discontinuity surface. The ratios of the of mean numbers
of model particles of every species in one cell ahead of and behind the discontinuity surface (Na and Nb respectively)
were defined by the discontinuity condition. (Here and below indices a and b refer to variables ahead of and behind
the shock wave.)
N aU a = N bU b ,
where Ua and Ub are the flow velocities ahead of the wave and behind it. The initial weighting factors for a given
species of particles in one of the regions ahead of or behind of the discontinuity surface were assumed to be equal,
and their values were selected in such a way that every component was represented by about the same number of
model particles.
During the evolution of the system, particles which pass the boundaries of flowfield were excluded from the
treatment. Fluxes of new particles into the modelled domain are specified in accordance with upstream and
downstream conditions. The position of the wave was stabilized by special procedure [3, 9]. It was assumed that the
amounts of substance, longitudinal momentum and longitudinal and transverse energy (r=1, 2, 3, 4 respectively)
transported by the particles of the type being considered across the boundaries were denoted by Zφ r (φ =a
corresponds to the left-hand boundary and φ =b corresponds to the right-hand boundary). Here and below, where
possible, the index indicating the type of the particle will be omitted for simplicity. Functions µφ have been
constructed, so that statistically equitable combinations of Zφ r occur in them and µφ→0 when t→∞ for the specified
conditions on the boundaries. In order to do this, the theoretical values of expectations of Zφ r , i.e. Eφ r(Zφ r ) and their
dispersions Dφ r(Zφ r ) were used. In computing Eφ r and Dφ r, the numbers of model particles crossing the boundaries
in the positive and negative directions at instant t were assumed to follow the Poisson distribution. It is also assumed
that the flows on the boundaries of the simulation region satisfy the following conditions:
(i) gradient-free flow near the boundaries;
(ii) applicability of the hypothesis of molecular chaos;
(iii) statistical independence of ηm and cm;
(iv) verity, on average, of the following relations for ηm of model particles leaving the simulation region,<ηm+>,
or entering it again, <ηm->:
(η )
+ 2
m
η m+ = η m− ≡ η m ,
= (η m− ) ≡ η m2 .
2
The computations have yielded the following results:
Eφ 1 = m η m tU φ N φ / ∆x;
Eφ 2 = m η m t (U φ2 + Φ φ ) N φ / ∆x;
Eφ 3 = ( m / 2) η m tU φ (U φ2 + 3Φ φ ) N φ / ∆x;
Eφ 4 = 2Φ ϕ Eφ 1
Dφ 1 = m 2 η m2 t ( 2Φ φ )1 / 2 [exp( − Sφ2 ) / π 1 / 2 + Sφ erf ( Sφ )]N φ / ∆x;
Dφ 2 = m 2 η m2 t ( 2Φ φ ) 3 / 2 [(1 + Sφ2 ) exp( − Sφ2 ) / π 1 / 2 + Sφ (1.5 + Sφ2 )erf ( Sφ )]N φ / ∆x;
N
m2 2
η m t (2Φ φ ) 5 / 2 [( 2 + 4.5Sφ2 + Sφ4 ) exp( − Sφ2 ) / π 1 / 2 + Sφ (3.75 + 5Sφ2 + Sφ4 )erf ( Sφ )] φ ;
∆x
4
2
= 8Φ φ Dφ 1 .
Dφ 3 =
Dφ 4
Here
Φ φ = kT / m,
Sφ = U φ /(2Φ φ )
1/ 2
,
erf ( x ) = ( 2 / π
x
1/ 2
) ∫ exp( − y 2 )dy.
0
The functions µφ have the form
4
µφ = ∑ bφ r [( Z φ r − Eφ r ) / Eφ r ]2 .
r =1
Here bφ1=1, bφ r=(Eφ r/Eφ 1)2 Dφ 1/ Dφ r , r = 2, 3, 4.
The procedure being considered was carried out after phase A and consisted of the continuous introduction of
particles of component with the greatest concentration until the entering particles reduced µφ.
3 RESULTS OF SIMULATION IN N2
The simulation of shock wave in N2 was carried out for Max number M=1.71, temperature Ta=200K in order to
test used model of rotational relaxation at results of experiments of [10]. 28 rotational states, RR and RT-transitions
were taken into account as described in section 2. RT-transitions with change of rotational states j up to 12 were
considered. Mean values of parameters of stream were obtained using number of sampling NS=200, Na=288.
A good agreement with experiments of [10] was obtained for R=0.07 when for l’>p’ Bp’=0.05 by 0≤p’<7, Bp’=0.5
by 7≤ p’<17, Bp’=1 by 17≤ p’< 27 and when for l’< p’ Bp’=1 by 0≤p’≤ 27 (see equation (1)). The reasons of
introducing of Bp’ different from 1 are the use of σlm,pq which was destined for the case of collisions of diatomic
molecule and molecule of noble gas and for the case of the model of collisions which was not the model of hard
spheres. (It should be noted that the model of hard spheres was used in considered simulation.) All this may break
the principle of detailed equilibrium.
10 - 1
1.0
fr
no,Tio
4
10 - 2
3
3
1
10 - 3
2
0.5
1
10 - 4
4
2
10 - 5
0.0
-10
0
x/λ 10
FIGURE 1. Profiles of relative concentration
and temperatures of N2.
0
5
10
15
20
25
j
FIGURE 2. Rotational distributions of N2.
Figure 1 shows the simulated profiles of relative concentration no=(n-na)/(nb-na) (curve 1) and relative kinetic
temperatures Tio=(Ti-Ta)/(Tb-Ta) (curve 2 - rotational temperature Tro, 3 - total temperature Tso). Experimental values
of no and Tro [10] are also shown in Fig. 1 as circles and triangles.
Figure 2 shows the rotational distributions fr of molecules in the front of wave. Curves 1 and 4 are equilibrium fr
ahead of and behind the wave. Squares and rhombi are distributions obtained at the left and right sides of the
modeled domain. Curves 2-3 are the fr in the front: 2 – x = - 3,08: 3 – x = - 0.08. As one can see, fr are near
Boltzmann distributions at the boundaries of the modeled domain. The relaxation occurs monotonously without
explicit peculiarities.
nio, Tio
10-1
6
1.0
1
fr
5
2
3
3
10-5
7
4
4
10-3
0.5
10-7
2
10-9
0.0
-10
-5
0
5
10 x/ λ
FIGURE 3. Profiles of relative concentration and tem-
1
0
10
20 30
40
50 j
FIGURE 4. Rotational distributions of O2, 0≤j≤57.
peratures for mixture of He, O2 and Xe, 0≤j≤57.
The simulation was repeated, but RT-transitions with change of j up to only 6 were considered. In this case
R=0.1 and for l’>p’ Bp’=0.1 by 0≤p’<7, Bp’=0.7 by 7≤p’<17, Bp’=1 by 17≤p’< 27 and for l’<p’ Bp’=1 by 0≤p’≤27
Obtained profiles of no and Tso are very close to that in the previous calculation. The slope of the profile of Tro is
more less (see curve 4 in Figure 1). The agreement with experiments [10] was not so good as above.
A performed simulations without consideration of RR-transitions have given practically the same results as
above.
4 RESULTS OF SIMULATION FOR MIXTURE OF HE, O2 AND XE
The simulation of a shock wave was carried out for the case of gas mixture of He, O2 and Xe with numerical
density ratio 200:1:1 at M=4 and Ta=293, NS=750, Na=120. It was carried out taking into account rotational
relaxation of O2. There were simulated RR and RT-transitions as described in section 2. RT-transitions with change
of j up to 12 were considered.
Initially, 58 rotational levels were taken into account. As in this case of N2, R was equal to 0.07 and for l’ >p’
Bp’= 0.05 by 0 ≤ p’<7, Bp’= 0.5 by 7≤p’<17, Bp’= 1 by 17≤ p’< 57 and for l’< p’ Bp’ =1 by 0 ≤ p’≤ 57.
Figure 3 shows the simulated profiles of relative concentrations nio=(ni-nia)/(nib-nia) (curve 1 – He, 2 – O2, 3 –Xe)
and relative kinetic temperatures Tio=(Ti-Ta)/(Tb-Ta) (curves 4 – 6 – total temperatures of He, O2 and Xe,
respectively, 7 – Tro of O2). The behaviour of concentrations and total temperatures resembles that for the case
without consideration of rotational relaxation except temperature of O2.There is no excess of the temperature of O2
in front over its equilibrium value behind the front. Profile rotational temperatures of O2 is analogous to that for the
case of N2.
Figure 4 shows the rotational distributions fr of molecules of O2 in the front of wave. Curves 1 and 4 are
equilibrium fr ahead of and behind the wave. Squares and rhombi are distributions obtained at the left and right sides
of the modeled domain. Curves 2, 3 are the fr in the front: 2 – x = -7.58, 3 – x = - 4.58. As one can see, fr are near
Boltzmann distributions at the boundaries of the modeled domain except the lowest rotational states at the right
boundary. Relaxation occurs monotonously without explicit peculiarities like in the case of N2.
10-1
6
o
G23
ni ,Ti
5
3
10-4
o
1
1.0
2 7
4
10-7
5
3
2
4
1
10-10
10-13
0.5
6
0
2
4
6
g/a
FIGURE 5. Relative velocity distributions, 0≤j≤57.
0.0
-10
-5
0
5
x/λ
10
FIGURE 6. Profiles of relative concentration and tem-
peratures for mixture of He, O2 and Xe, 0≤j≤77.
Figure 5 shows the relative velocity distributions G23 for pairs containing one molecule of O2 and one molecule
of Xe. Velocities g are normalized to the velocity of sound a in the gas ahead of the wave. Curves 1 and 6 are
equilibrium distributions ahead of and behind the wave. Crosses and oblique crosses are distributions obtained at the
left and right sides of the modeled domain. Curves 2-5 are the distributions in the front obtained at x= - 7.58, - 5.32,
- 4.58 and - 3.08, respectively. Though the consideration of rotational relaxation of O2 has led to 25-fold decrease of
the maximum superequilibrium at g = 6.75 corresponding to Ed (MSE) it remained as previously considerably high
(MSE=4·104). Deviation of G23 from equilibrium meanings at the left boundary of the modeled domain is caused by
influence of shock wave. An increase in the modeled domain must decrease the deviation down to zero.
It should be noted that component O2 is represented by greater number of model particles than other gases. The
ratio of the number of model particles of O2 and of that of another component is approximately equal to the number
of considered rotational states of O2. So, obtained velocity distributions for O2 and for pairs of molecules containing
O2 reached lower meanings than other distributions because in these cases samplings were more rich.
It should also be noted that, as in the case without consideration of rotational relaxation, the distributions of
relative velocities g for pairs of molecules with Xe substantially exceeded in the front their equilibrium values
behind the wave at high values of g. Though special efforts were not directed on investigations in this field but
obtained values of maximum reached superequilibrium for distributions of pairs of molecules Xe-Xe and Xe-He
were about 100. Samplings for these distributions were not so rich as in [5, 6].
Next, another analogous simulation was carried out. The difference consisted in that R=0.1 and all Bp’ =1. In this
case profiles of nio and Tio coincided practically with that shown in Fig. 3 except profile of Tro for O2. Profile of Tro
achieved only meaning 0.9 at the right boundary of the modeled domain . Obtained fr at the left and right sides of the
modeled domain were equilibrium distributions with good accuracy. Probably this is due to the fact that σlm,pq for
RT-transitions were used without distortions, most of RT-transitions were results of collisions of diatomic molecules
with molecules of noble gas and the role of RR-transitions was negligible. Indeed, in this case obtained in the
simulation a number of RR-transitions was very low in comparison with the number of RT-transitions. (The last
took place in all cases under consideration in this section.) Obtained value of MSE was equal to 6·104.
At right boundary of modeled domain, the maximum value of fr was only a slightly higher than the value of fr for
taking into account highest j in all cases considered above in this section. The ratio of these values frm/frh was
approximately equal to 10. Probably, this was the reason why the value of Tro did not achieve meaning 1 at the right
boundary of the modeled domain. We should expect that increase of the number of rotational levels taking into
consideration will decrease this deviation.
Therefore, simulation was carried out in which 78 rotational levels were taken into account, R=0.1 and all Bp’=1.
Figure 6 shows the simulated profiles of nio and Tio. Denominations are the same as in Fig. 3. As one can see, profile
of Tro achieved level 1 at the right boundary of modeled domain. And obtained fr at the left and right sides of the
modeled domain were equilibrium distributions with good accuracy (see Fig. 7, denominations are the same as in
Fig. 4, 2 – x = -7.58, 3 – x = - 4.58). Moreover, the ratio frm/frh was approximately equal to 100. Obtained value of
MSE was equal to 2·105. (See Fig. 8, denominations are the same as in Fig. 5, 2 – x = -7.58, 3 –x =-5.32, 4 –x =-4.58
and 5 – x= -3.08.) This simulation was repeated with R=0.07 and R=0.03. Obtained value of MSE was equal to 105
for R=0.07 and 2·105 for R=0.03.It should be noted that slope of the profile of Tro becomes more slight with decrease
of R. A profile of Tro is less slight from the meaning 1 at the right boundary of the modeled domain for R=0.03.
10-1
fr
10-1
G23
4
-4
10
10-7
10-4
3
3
10-7
10-10
-13
10
2
10-10
5
1
2
10-13
10-16
4
1
0 10 20 30 40 50 60 70 j
FIGURE 7. Rotational distributions of O2, 0≤j≤77.
6
0
2
4
6
g/a
FIGURE 8. Relative velocity distributions, 0≤j≤77.
5 CONCLUSION
In order to take into account rotational relaxation of diatomic molecules in the front of a shock wave, the
modified Monte Carlo method of unstationary statistical simulation with variable weighting factors was used.
Setting of sufficient number of rotational levels and reasonable cross sections of rotational transitions allowed to
take into account quantum-mechanical nature of the rotational relaxation. It was impossible without use of
weighting factors because the ratio of maximum value of fr and the value of fr for taking into account highest j
(frm/frh) achieved 1017.
The method checked on the shock wave in N2 was used for simulation of shock wave in gas mixture of He, O2 and
Xe. Although, for slightly different models of rotational relaxation, results of simulation have shown maximum 25fold and minimum 5-fold decrease of the effect of superequilibrium for G23 the value of the effect was still high.
Consequently, the dissociation of O2 in one step by collisions with Xe may have a considerably higher rate in the
front than the total dissociation behind the wave. And some amount of atoms of O may be formed in the front of a
shock wave. Introduction of small amount of H2 in the considered mixtures of He, O2 and Xe can not substantially
affect the relaxation in the front. And the formation of atoms of O in the front may decrease a period of the
induction of the interaction of oxygen and hydrogen behind the shock wave.
The increase in required computational time due to consideration of rotational levels of O2 as separate
components is compensated by more rich sampling for G23.
ACKNOWLEDGMENTS
The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 97
-03-32481a).
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