The calculation of TV, VT, VV, VV’ ? rate coeﬀicients for the collisions of the main atmospheric components

. The ﬁrst-order perturbation approximation is applied to calculate the rate coe(cid:129)cients of vibrational energy transfer in collisions involving vibrationally excited molecules in the absence of non-adiabatic transitions. The factors of molecular attraction, oscillator frequency change, anharmonicity, 3-dimensionality and quasiclassical motion have been taken into account in the approximation. The analytical expressions presented have been normalized on experimental data of VT-relaxation times in N 2 and O 2 to obtain the steric factors and the extent of repulsive exchange potentials in collisions N 2 -N 2 and O 2 -O 2 . The approach was applied to calculate the rate coe(cid:129)cients of vibrational-vibra-tional energy transfer in the collisions N 2 -N 2 , O 2 -O 2 and N 2 -O 2 . It is shown that there is good agreement between our calculations and experimental data for all cases of energy transfer considered.


Introduction
Vibrational excitation of atmospheric molecules is important for the thermal structure of the atmosphere and may signi®cantly change the chemical structure through modi®cation of the normal reaction rates.Whereas the rotational and translational modes of the molecules with short relaxation times may be equilibrated during atmospheric disturbances (auroral precipitation, arti®cial heating etc.), the vibrational degrees of freedom will de®nitely not be equilibrated, thus enhancing the chemical activity of atmospheric components.
The reactions involving vibrationally excited molecules are believed to be faster than the corresponding ground level reactions, because they occur on potential energy surfaces which have little or no activation barriers and are highly exothermic (Rusanov and Fridman, 1984).When the vibrational temperature of the molecular atmospheric components is suciently enhanced, the rates of reactions O N 2 3 NO N 1 may be greatly increased causing eective NO production in thin layers during auroral beam-plasma instabilities (Mishin et al., 1989;Aladjev andKirillov, 1995, 1997).The eect of selective reactant excitation on the rates of chemical reactions has been shown by Eyring et al. (1980), andSmith (1980).With an activation barrier displaced into the ``exit valley'' of the potential, vibrational energy was again found to promote reaction much more eectively than relative translational energy.Also vibrationally excited molecules play a signi®cant role in the ionic chemistry and thermal balance of the upper atmosphere.For example, the chemistry of ion O in F-region of the ionosphere is very dependent on the vibrational temperature of ionospheric components.Nonlinear theory of the production of main ionospheric maximum was developed by Richards and Torr (1986), and Vlasov and Izakova (1989).It was found that enhanced vibrational excitation of atmospheric components results in the increase of O losses and the decrease of electron concentration.The collisions of excited N 2 with thermal ionospheric electrons promote the heating of the electron gas and, under conditions of enhanced F region electron densities, N 2 may act as a small net source of electron thermal energy (Richards et al., 1986).
An understanding of rates of dierent plasma-chemical processes involving excited molecules helps to explain the role of vibrationally excited particles in the balances of ionospheric plasma because the ionospheric components can be excited to high vibrational levels during natural and arti®cial atmospheric disturbances.The classical, semiclassical, quasiclassical and exact quantum mechanical models developed to calculate the energy transfer in nonreactive vibrational-translational and vibrational-vibrational molecular collisions have been reviewed in Rapp and Kassal (1969), Nikitin (1974a, b), Nikitin and Osipov (1977) and Billing (1986).The purpose of this study is to show that the ®rst-order perturbation approximation (FOPA) of the calculation of energy transfer between molecules in the absence of non-adiabatic transitions gives the TV, VT, VV and VV H -coecients corresponding to the experimental data if the factors of molecular anharmonicity, frequency shift, attraction etc. are included.The present method is applicable to transitions involving the exchange of one or more quanta.Application is made to the N 2 -N 2 , O 2 -O 2 , N 2 -O 2 molecular collisions.

Vibrational-translational energy transfer (TV and VT-process)
The basic model for the investigation of vibrationaltranslational energy transfer is the collinear collision between a oscillator and an incident particle interacting through a repulsive exponential potential.Let us consider the collinear collision of harmonic oscillator AB and a structureless particle C, shown in Fig. 1.The harmonic oscillator AB has the classical oscillator frequency x, and the repulsive interaction potential between the atom C and the nearest atom of the harmonic oscillator, B, is assumed to be exponential.The Hamiltonian of the interaction is equal to: For calculational purposes, it is convenient to de-®ne the dimensionless quantities z a z 0 , x ar À r e wal 1a2 , s xt (Nikitin, 1974b).Then the Hamiltonian Equation (3) becomes in the classical case where i 0 lx 2 aa 2 Y expz 0 akr e i 0 ae 0 Y m k 2 law.The equations of translational and vibrational motions for the system can be shown to be: x 0 X 6 Since x ( 1, one can perform an approximate integration of Eq. ( 5) and ( 6) neglecting x in the exponential term.The result of the integration is Here it is suggested that the oscillator was not excited originally, i z a 2 m 2 a2x 2 is the energy of relative motion on in®nite distance at s ÀI for unity reduced mass of AB and C, n xaam 2i z À1a2 ) 1 is the Massey factor.The expression (8) is identical with the energy amount which can be obtained from an approximate semiclassical calculation (Rapp and Kassal, 1969).In the semiclassical calculation, the molecule AB is treated as a quantum mechanical system with discrete vibrational levels.The relative translational motion of the molecule and the particle C is treated classically, see Eq. ( 7).
The possibility of vibrational-translational energy transfer (TV or VT-process from vibrational level n to m) calculated according to the semiclassical FOPA is equal to where " h is Planck constant, " hx nm ji n À i m jY i n and i m are the energies of vibrational levels n and m, p t is the power acting on the oscillator, r nm W n r À r e W m dr Y 10 For the harmonic oscillator jr nYn1 j 2 n 1jr 01 j 2 n 1 " h 2wx 01 and the possibility of n 6 n 1 energy transfer is proportional to that of 0 6 1: It can be shown that the energy calculated according to Eq. ( 8) is related to the semiclassical possibility: Di gv g 01 " hx 01 X Kelley and Wolfsberg (1966) have calculated the classical energy transfer to the oscillator by two methods.The approximate procedure neglected the eect of the oscillator motion on the external collisional motion in coordinate .The exact procedure has taken into account the in¯uence of the oscillator motion on the external collisional motion.It was found that the ratio of approximate and exact energy transfers is not equal to unity and depends only on m: Secrest and Johnson (1966) presented an exact quantum-mechanical solution for vibrational-translational transition probabilities in the collinear collision of a particle with a harmonic oscillator.They have found that their results did not reduce to the ®rst-order distorted-wave approximation (FODWA) and the probabilities of Secrest and Johnson (1966) appear to be proportional (but not equal) to the probabilities of FODWA.Rapp and Kassal (1969) believed that the reason for the failure of the FODWA is intimately related to the failure of the approximate classical calculations of Kelley and Wolfsberg (1966).This is the quantummechanical analogy of the classical approximate procedure, because the wave function in coordinate r is totally independent of coordinate .Therefore, the FODWA must fail to agree with the exact quantum-mechanical calculations to the same extent that the approximate classical calculations fail to agree with the exact classical calculations.
The factor (12) may be regarded as a correction factor for the usual FODWA transition probabilities and approximate classical calculations.It is also interesting to note that the revised FODWA given by Mies (1964) leads to a correction factor expressible as 1 À m F F F.
As was pointed out in Nikitin (1974a, b), Nikitin and Osipov (1977) and Nikitin et al. (1989), the disagreement of the approximate calculation of energy transfer with the exact one is reduced in the classical case if the frequency shift factor is taken into account.Let us consider the approximate equation of oscillator motion with the frequency shift (obtained from the Hamiltonian Equation ( 4)): in which the place of potential energy minimum x 0 is determined by the expression x 0 0 X 14 The comparison of TV-energy transfer for the case of oscillator frequency change with the expression (8) (Nikitin 1974a, b;Nikitin and Osipov, 1977;Nikitin et al., 1989) gives the factor Here t 1 is the Bessel function.When the mass factor m is less than about 1a2, the approximate formula may be used.
The formula ( 16) can be obtained if Green's method is applied to calculate the net amount of vibrational energy transferred to the oscillator (Kirillov, 1997).Green's function of the Eq. ( 13) is equal to Baz et al. (1971): where the solution of homogeneous Eq. ( 12) is The integration of the product of the function on the right side of (13) and Green's function (17) leads to the factor (16) (Kirillov, 1997).
To obtain the averaged possibility of vibrationaltranslational energy transfer m 0 6 m 1, the expression (8) must be integrated over a normalized Maxwell distribution of relative collision velocities along the line of centres.The resulting averaged transition probability is where the vibrational and translational factors m and tr are as follows: Here k and are the Boltzmann constant and the translational temperature.The calculation of vibrational-translational energy transfer could be easily extended to the case of an anharmonic oscillator by replacing the harmonic AB wave functions by anharmonic wave functions where the frequency for the transfer n 6 n 1 is equal to where x e is the anharmonic constant.Taking into account the frequency reduction in (21) and the dependence of translational and vibrational factors tr and m on the frequency of the transfer (20a, b), one can obtain the correction factor in the n 6 n 1 energy transfer for the anharmonic oscillator: The expression (20b) was obtained for the repulsive exponential potential.In fact, there is an eect of a longrange attractive potential on the energy transfer.A procedure for the determination of the eect of attraction was suggested in Nikitin (1974a).To obtain the factor of this attraction we have used the relation between interaction times s HH , s H with and without the attraction, respectively, following Nikitin (1974a): where i is the relative energy of the molecule and the particle at in®nite separation before collision and e is the depth of the potential well.Place the Eq. ( 23) into the expression for a probability of oscillator excitation, (proportional to expÀ2pn in adiabatic collisions according to Eq. ( 8)), and average over the Maxwell distribution.The integrated function in this case is Nikitin (1974a): where The function (24) has its maximum near the exponential extremum and the integration over the Maxwell distribution yields the factor expÀ3c 2a3 0 2gc 1a3 0 , where c 0 y 3a2 0 (Kirillov, 1997).So the factor of the attraction is equal to: The exponential power in Eq. ( 25) is p 2 greater than the one obtained in Nikitin (1974a).
Although the one-dimensional collision study leads to a functional form of the transition probabilities, it is necessary to estimate the possibility of vibrationaltranslational energy transfer for 3-dimensional case, as made by Schwartz and Herzfeld (1954), Calvert andAmme (1966), andHansen andPearson (1970).The three-dimensional model is needed to ®t both the slope and the absolute magnitude of the experimental relaxation rates to our theoretical estimations.In this study the treatment of 1-dimensional calculation is extended to a collision in three dimensions by introducing the steric factor st .
The ®rst eect of the 3-dimensionality results from the necessity of taking into account the contributions of non-collinear collisions.Since the collision of the molecule AB with the particle C may not be linear, a consideration of the eect becomes necessary.The second important eect accounted for by the factor st is caused by the non-zero-impact parameter collisions.It is obvious that an accurate prediction of the value of the steric factor st is practically impossible.
If the vibration is treated as if it is a breathing vibration in a nearly spherical molecule, one may expect for the diatomic molecule, that the value st for vibrational-translational transition probability is equal to hos 2 hi 1a3 where h is the angle between molecular axis and (Schwartz and Herzfeld, 1954;Billing, 1986).The conclusion is based on the suggestion of very weak interaction at angles h $ pa2.Quantum-mechanical calculations of the real potential surfaces show stronger angular interaction at h $ pa2 than in the case of the pairwise potential (Nikitin et al., 1989).So, in fact, one may expect greater magnitudes of st .
Thus, the transition probability for the binary collision in which target molecule AB undergoes a VT or TV-transition n 6 n 1 upon the impact of an incident particle C is as follow: where all factors (11), ( 15), ( 20a), ( 20b), ( 22), ( 25) have been taken into account and the value of quasiclassical factor can be obtained in the case of quasiclassical consideration (Landau and Lifshitz, 1965;Nikitin, 1974a).Signs and À are to be used for VT and TV-processes, respectively.The rate coecient of vibrational-translational energy transfer n 6 n 1 can be described by the equation: where 0 is the gas-kinetic radius of the collision.

Vibrational-vibrational energy transfer (VV and VV H -process)
The simple FOPA calculation can be extended to the case of vibrational-vibrational energy transfer between molecules.Let us consider two diatomic molecules AB and CD colliding head-on (Fig. 2).As in a previous section, the intermolecular potential is assumed to be purely repulsive between B and C, in analogy to previous calculation of vibrational-translational energy exchange.The Hamiltonian of the interaction is equal to in which the distance between centres of mass of AB and CD is , the respective oscillator coordinates in AB and CD are r 1 and r 2 Y r 1e and r 2e are the equilibrium separations in AB and CD, Y p r1 Y p r2 are the conjugate impulses, As in the case of vibrational-translational calculation we suggest that the relative velocity of AB and CD at t ÀI has the value m, but the oscillator CD is initially vibrationally excited and the energy of the vibration is Using the dimensionless quantities z a z 0 Y x 1 ar À r 1e w 1 al 1a2 Y s x 1 t, we obtain the following approximate expression for the Hamiltonian (29) in the classical case: where we neglect the energy loss of the CD oscillator and use for the amplitude of the CD vibration k 2 a2" haw 2 x 2 1a2 f ( 1, and we have i 0 lx 2 1 aa 2 Y expz 0 ak 1 r 1e k 2 r 2e i 0 ae 0 Y m k 2 1 lawY d is the phase of CD oscillation at t 0.
The equation of vibrational motion of the AB molecule is and taking into account the expression for translational motion ( 7) the vibrational-vibrational energy transfer can be obtained as where i z a 2 m 2 a2x 2 1 and n x 2 À x 1 aam.The energy amount in Eq. ( 32) can be calculated according to semiclassical FOPA.The approximation gives the following expression for VV and VV¢ energy transfer: where i n Y i m are the vibrational energies of AB and CD before the collision and i n H Y i m H are the ones after the collision.The calculation of the integral in Eq. ( 33) leads to the expression Here the vibrational factors are As in the case of VT-energy transfer there is the relation of classical and semiclassical calculations 10g 01 " hx ef 01 mf 2 i z pn Á cschpn 2 i 0 % Di gv X To obtain the factor of frequency shift for VV and VV H -energy transfer one has to consider Eq. ( 13) but here the minimum of potential surface is determined from the equation Green's function of Eq. ( 13) is similar the one calculated according to Eqs. (19±21), since the amplitude of CD oscillations in small B ( 1.The integration of Eq. ( 13) for vibrational-vibrational energy transfer leads to the factor ( 16).
The integration of Eq. ( 34) over the Maxwell distribution gives the expression The function p c is expressed as (Keck and Carrier, 1965): for the quasiresonant transfer c % 1 and for a very large resonance defect c ) 1.
The factor of the attraction of the molecules can be obtained as in the case of VT-calculation.The interaction time according to Eq. ( 23) has to be inserted in the integral (36b).For a very large resonance defect c ) 1 the factor is and for a small resonance defect c % 0 the factor is Thus, the FOPA transition probability for the binary collision in which target molecule AB undergoes a VV or VV H -transition n 3 n H upon the impact of an vibrationally excited molecule CD is as follows: where all factors ( 16), (36a), (36b), (37a), (37b), (38a), (38b) have been taken into account.The value of the quasiclassical factor is equal to where signs and À are to be used for excess and defect energy VV or VV H -processes, respectively.It is suggested that the contribution of the dierent orientations of the AB and CD molecules has been included in the multiplying of steric factors in Eq. ( 39).For the case of a breathing vibration in a nearly spherical molecule, the averaged value of the multiplying for the vibrational-vibrational transition probability is equal to hcos 2 h 1 ihcos 2 h 2 i 1a9, where h 1 and h 2 are the angles between molecular axes and (Billing, 1986).The in¯uence of the anharmonicity will be taken into account in the calculations of the factors f sh Y f qu Y p c and vibrational factors (35), which depend on the frequencies of the vibrations.
The rate coecient of vibrational-vibrational energy transfer can be obtained as in the case of TV, VT-transfer (28), i.e., by multiplying the probability (39) by the averaged thermal velocity and gas-kinetic cross section.

Results of the calculation and the comparison with experimental data
The expression (28) has been normalized on the experimental data of VT-relaxation time in N 2 and O 2 from (Millikan and White, 1963;Zabelinskii et al., 1985) to obtain the magnitudes of steric factors and the extent of repulsive exchange potentials.The magnitudes of e are taken from Radzig and Smirnov (1980) and the gaskinetic radii of N 2 and O 2 molecules from Polak et al. (1973) andCamac (1961).We use the relation of VT-relaxation time with the rate coecient according to Nikitin (1974a).
The interest in the study of relaxation processes in gases has resulted in the accumulation of a large body of experimental data concerning the rate constants of vibrational energy transfer in molecular collisions.In particular this concerns the rate coecients of VV and VV H -processes in the collisions of the main atmospheric components N 2 and O 2 .In accordance with the formula (39) we have calculated the rate coecients of VV and VV H -energy transfers for the collisions N 2 -N 2 , O 2 -O 2 , N 2 -O 2 .The factor of attraction has been taken into account according to Eq. (38a) for c b pa4 3a2 and Eq.(38b) for c `pa4 3a2 .
In the calculation of the rate coecient for the resonant process we have used a N 2 ÀN 2 39 nm À1 .The formula obtained for the process (41) is the following: The temperature dependence of Eq. ( 42) is shown in Fig. 3.Here the experimental estimations of Suchkov Fig. 3.The temperature dependence of the rate coecient for the process N 2 1 N 2 0 3 N 2 1 N 2 0: dashed line; our calculation: solid line; semiclassical trajectory calculation of Billing and Fisher (1979), Billing (1986), experimental data of Suchkov and Shebeko (1981) and Shebeko (1981), Akishev et al. (1982), Valyanskii et al. (1984a, b), and Gordeev and Shahatov (1995) for the rate coecient are presented.There is good agreement of the formula with experimental data of Suchkov andShebeko (1981), andValyanskii et al. (1984a, b) and some excess over the estimations of Akishev et al. (1982) and Gordeev and Shahatov (1995).Also the semiclassical trajectory calculation of Billing and Fisher (1979) and Billing (1986) is presented in Fig. 3 showing good agreement with two smaller experimental estimations.
In the calculation of the rate coecient for the process we have used a O 2 ÀO 2 45 nm À1 .The formula obtained for the process ( 43) is the following: The dependence of the sum of rate coecients of VVprocesses ( 43) and VT-processes on vibrational levels n at temperature 300 K is compared with experimental data of Park and Slanger (1994) in Fig. 4. Also the semiclassical trajectory calculations of Billing and Kolesnick (1992) are presented here.
To calculate the rate coecients for N 2 -O 2 collisions we suppose from Billing (1994) The formula means that the radius of the overall exchange potential is the sum of short range potential radii of colliding molecules (Nikitin et al., 1989).So the calculated rate coecient is for the VV H -process for the other VV H -process where in the calculation of vibrational factor we have used a formula for the matrix element of multiquantum transfer obtained by Herman and Shuler (1953) for Morse potential approximation.The results of the calculation according to Eqs. ( 46) and ( 48) are compared with the experimental data of Gilmore et al. (1969) and Park and Slanger (1994) in Figs. 5 and 6.
There is seen good agreement of the calculations and experimental data for both kinds of VV H -energy transfer in the collision of molecular nitrogen and oxygen.The semiclassical trajectory calculation of Billing (1994) for the process ( 49) is also presented in Fig. 6.As in the case of N 2 -N 2 collision there is some overestimation by our calculations of the trajectory results.The large dierence of our and trajectory calculations can be explained not only by the dierent approximate methods.The theoretical calculations for VV-exchange in molecular nitrogen using the integral quasiclassical representation method (Zhuk and Klopovsky, 1988) have shown the sensitivity of the results to chosen potential surface.As can be seen from their calculation, the rate constants obtained with dierent potential surfaces dier by factors of 5±7 but the temperature dependence is similar.

Conclusions
The FOPA was one of the ®rst methods to have been applied in the calculation of vibrational energy transfer in the collisions involving vibrationally excited molecules.The simplicity of analytical expressions for the rate coecients of vibrational-translational and vibrational-vibrational processes has allowed us to use eectively the results of the FOPA calculation in an interpretation of experimental data of the vibrational energy relaxation.But sometimes the disagreement of theoretical estimations based on the approximation with the results of experimental measurement has led to the suggestion that the application of the simple method is not correct in the calculation of the rates of vibrational energy transfer processes.Nikitin (1974a, b), Nikitin and Osipov (1977), Nikitin et al. (1989) have pointed out some factors of atommolecular collision which are to be taken into account in the calculations according to the FOPA.We have used the FOPA to obtain the analytical expressions of the rate coecients of TV, VT, VV and VV H -energy transfer in molecular collisions of the main atmospheric compo-nents.The factors of molecular attraction, oscillator frequency change, anharmonicity, 3-dimensionality and quasiclassical motion have been considered in the approximation.We have normalized the presented analytical expressions on the experimental data of VTrelaxation times in N 2 and O 2 to obtain the steric factors and the extents of repulsive exchange potentials in the collisions N 2 -N 2 and O 2 -O 2 .The obtained values of exchange potential radii are in good agreement with recent quantum-mechanical calculations.
The approach was applied to calculate the rate coecients of vibrational-vibrational energy transfer in the collisions N 2 -N 2 , O 2 -O 2 and N 2 -O 2 .It is shown that there is good agreement of our calculations with experimental data for all considered cases of the energy transfer.The disagreement with the results of semiclassical trajectory calculation can be explained both by the dierence of applied approximate methods and by the sensitivity of semiclassical trajectory calculation to the chosen potential surface.
the distance between C and the AB centre of mass, r, the oscillator coordinate, is the separation of A and B, r e is the equilibrium value of r, and p r are the conjugate impulses, w is the oscillator reduced mass m e m f am e m f , l is the reduced mass of C on AB, or m e m f m g am e m f m g , m e Y m f Y m g are the masses, respectively, of atoms A, B, C, e 0 and a are constants determining the amplitude and the range of intermolec-ular forces, k m e am e m f .We suggest that the relative velocity of AB and C at t ÀI has the value m.

Fig. 1 .
Fig. 1.The collinear collision of the molecule AB and a particle C