Two-modes approach to the state-to-state vibrational kinetics of CO2

Two-modes (symmetric–asymmetric) reduction of the CO2 vibrational kinetics has been introduced systematically for the purpose of state-to-state modeling. The two-modes approach is known from multi-temperature models. Its basic assumption is that vibrational energy of the CO2 molecule which depends on 4 quantum numbers can be approximated by a function of only 2 numbers. Subsequently, vibrational states of CO2 with the same quantum numbers of the effective symmetric and asymmetric mode can be grouped into one combined state. The paper presents the calculation of the average probabilities of transitions between combined states based on the Schwartz–Slawsky–Herzfeld theory. Closed analytical formulas are derived for vibrational–translational and vibrational–vibrational transitions.


Introduction
Conversion of CO 2 into CO in microwave gas discharges is extensively studied at present due to its potential signicance for production of synthetic fuels and electri cation of chemical industry [1]. It is expected that transitions between vibrational states of CO 2 play decisive role in the conversion processes [2,3]. The most straightforward and common way of numerical modeling of the complex multi-step chemical processes is the state-to-state kinetics which treats excited states as separate species. Within this approach the chemical system is described by a set of particle balance equations for each individual species-the master equations.
The CO 2 molecules have 3 modes of oscillations: symmetric stretching, symmetric bending and asymmetric stretching [4]. The double degenerate bending mode is a ro-vibrational mode which posses rotational moment. Subsequently, the vibrationally excited state is described by 4 quantum numbers. The total number of different vibrational states of CO 2 Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. with energies below the dissociation limit E diss = 5.5 eV is estimated to be ≈10 5 . Thus, the state-to-state approach in its direct form cannot be applied in practice even to 0D problems of the CO 2 plasma conversion because of too high computational costs.
One possible solution of this issue is offered by the model proposed in [5][6][7] where vibrational states are combined into effective asymmetric states with equilibrium (Boltzmann) distribution of symmetric modes. That is, only 21 vibrationally excited CO 2 species have to be taken into account 1 . This model has already found wide application and serves as a basis for creating models with further reduced number of species suitable for 2D/3D ow calculations. Such as the reductions based on the principal components analysis [8] and on combining the asymmetric levels into 'lumped states' [9]. Automatic algorithms of grouping the energy states into bins are developed as well [10]. Further optimization of the computational performance is achieved in the models with assumed shape of the vibrational distribution function [11] and with continuum energy spectrum approximations. Examples of those lat-ter are diffusion equation in the vibrational energy space [12] and the recently proposed method based on the Fokker-Plank equation [13,14].
Nevertheless, the computationally fast reduced and continuum spectrum models rely on the detailed state-to-state calculations which serve as benchmarks for veri cation and calibration. The state-to-state approach remains a tool which can revive internal mechanisms of vibrational kinetics otherwise not accessible by other methods.
A serious drawback of all the models of the CO 2 chemical conversion process listed above-starting from [5]-is that they focus on the kinetics of asymmetric modes. The nonequilibrium dynamics of the symmetric modes is neglected on assumption of their fast thermal equilibration. This assumption is, however, cannot be veri ed by the models themselves.
Both symmetric and the asymmetric vibrational modes of CO 2 are taken into account in the state-to-state model [15] developed for the spacecraft atmospheric entry simulations. In this model the ro-vibrational bending modes are combined into one effective state described by 3 quantum numbers instead of 4. This yields ≈ 10 4 model species with vibrational energies smaller than E diss . Solving a set of master equations of that size turned out to be still too computationally demanding. In practical applications of that model the number of effective states had to be reduced down to 1224 (states with vibrational energy smaller than 3 eV) [15,16].
It is possible to devise a model which can be placed in the middle between the model of [5,6] and the one of [15,16]. The vibrational energy of CO 2 can be approximated as being a function of only two numbers-the mode numbers of symmetric (v s ) and asymmetric (v a ) vibrations [3]. This two-modes approximation which goes back to the multi-temperature models of the CO 2 vibrational spectrum [2,3] allows to combine the states with same v s , v a into one effective state. The total number of such 'combined states' with vibrational energies below E diss is around 900. That is, the state-to-state description in this approximation is computationally feasible at least in 0D/1D simulations. At the same time, the assumption of thermal equilibrium is applied here only within the groups of symmetric vibrations with close energies. The populations of such groups represented by the combined states are calculated by the model itself. In particular, the pump up of vibrational energy due to vibrational-vibrational exchange between the symmetric modes can be investigated. This is the main difference between the two-modes approximation and the model of [5,6].
The two-modes state-to-state model is technically not difcult to implement in a computer code. The main dif culty was found to be the absence of readily available formulas for the probabilities of transitions between the combined symmetric-asymmetric states. The present paper intends to close this gap. The calculation of those transition probabilities is the main subject of this paper. Same as in [5,6,15,16] the consideration is based on the Schwartz-Slawsky-Herzfeld (SSH) theory [17,18,20]. This is an approximate rst order perturbation theory based on the assumption that the shortrange interaction dominates. This model gained wide popularity because it leads to simple analytic expressions for transition probabilities. At the same time, the model was found to be suf ciently accurate for vibrational-translational processes [19,20]. Moreover, the rate coef cients [21] calculated using the SSH formulas as scaling relations were recently applied in [22] for modeling the CO 2 kinetics in a glow discharge, and a good agreement with experiment was obtained.
The arguments above speak in favor of the SSH model that it can be used as a reasonable rst approximation which is valid at least qualitatively. It will be shown here that two extra assumptions lead to simple analytic formulas for the effective probabilities of transitions between combined states as well. Those are: (i) the linear oscillator approximation for calculation of matrix elements; (ii) the assumption that vibrational energy is exactly a function of v s , v a only. This latter assumption, when it holds, also allows not to take into account the Fermi resonance (between symmetric vibrations) explicitly.
The validity of those assumptions is expected to deteriorate for large v s , v a due to anharmonicity and subsequent smearing of the energy levels gathered into combined states. The picture of discrete levels can still be formally applied, but the transition probabilities, strictly speaking, may differ from those derived here. For high vibrational energies a near resonance collisionless energy exchange between symmetric and asymmetric modes leads to an effect similar to broadening of spectral lines which affects the probabilities of the vibrational-translational processes [12]. Also the justication used here for neglecting the Fermi resonance would not apply anymore. All those non-harmonic effects are very dif cult to quantify and this is not attempted in the present paper. This dif culty is not to last extent due to the lack of exact knowledge of the CO 2 vibrational energy dependence from the quantum numbers. A further brief discussion of this subject will be given in the conclusions.
Although the SSH theory is considered to be well known its individual pieces are scattered over a number of publications the reader might not be familiar with. Therefore, the paper starts from an overview and a summary of the theory in sections 2-4. Section 2 address the derivation of the basic equations. In section 3 the application of the SSH theory speci cally to CO 2 is described. The resulting formulas for vibrational transitions are summarized in section 4. Sections 5 and 6 present the original contribution of the present work. The two-modes approach and general equations for the probabilities of transitions between combined states are introduced in section 5. In section 6 selection of the most relevant processes is made, and formulas for the corresponding transition probabilities are derived. The last section concludes the paper and address some outstanding issues of the model.

Zener's method
The basis of the SSH model is the Zener's method [23]. In his original paper Zener considered a co-linear collision of a diatomic molecule BC with an atom A, gure 1. The system is described by the following wave equation: Subsequently, Ψ v are eigenfunctions of the molecular vibrations which correspond to vibrational energies (eigenvalues) E v , V m (x) is the energy of interaction of B and C as a function of separation x; µ m = m B m C / (m B + m C ). Zener proposed to look for the solution of (1) in the form: Substituting this equation into (1), multiplying the resulting equation by Ψ * u (x), integrating over x from −∞ to +∞, and making use of the orthogonality of Ψ v yields: here is the matrix element. Approximate solution of the equations (3) is found by the method of perturbations. The 0th order solution is an elastic collision where the initial state Ψ v 0 is not changed: Substituting this solution into (3) yields the equation for U v=v 0 : Approximation of the 1st order is obtained by substituting Q v=v 0 = U v 0 , Q v =v 0 = 0 into the right-hand sides of (3) which are then solved for Q v . The resulting solution has the following asymptotic form at r → +∞: The term describes an incoming wave-free particles which move toward each other, the term exp(i pv r) √ 2π p v , subsequently, an outcoming wave. See e.g. [24], section 21. That is, the rst term stands for the quantum state of the system before the encounter, the terms in the sum stand for the states after the encounter. According to the general rules of quantum mechanics, see e.g. [24], section 3, squares of the coefcients a u v then determine the probability of each state after the encounter w v→u which equals to [23]: Schwartz, Slawsky and Herzfeld [17] generalized the Zener's method to the co-linear collisions of poly-atomic molecules. For them the equation (2) is replaced by a more sophisticated wave equation written in terms of the normal coordinates of molecular oscillations. Here for clarity the particle A is still taken as a structureless object without oscillations. Generalization for the case when both colliding particles can change their vibrational states is straightforward and will be shown below in section 4. The interaction energy is in general a function V (x n , r) of all normal coordinates of the molecule x n and of the parameter r which characterizes the motion of the centers of masses of the colliding particles relative to each other. As this latter the distance between A and the closest atom of the molecule is usually taken, see section 3.2 below. Displacements of the oscillating atoms from their equilibrium positions can be assumed to be much smaller than r which allows to factorize the interaction term (an example will be shown in section 3.2): When (7) holds the integral (6) is separated as follows: here R u v is called 'the translationary part' of the transition probability, and V u v 2 is 'the oscillatory part'-square of the matrix element of the oscillating molecule; v stands for the values of its vibrational quantum numbers before collision, u-after collision. In this way the equation for molecular vibrations (2) and equation for the motion of the centers of masses (5) are completely separated from each other.
Equation (5) was solved analytically by Jackson and Mott for a special shape of the potential function V (r) = Ce −ar , where C and a are given constants [25]. The functions U v,u (r) which ful ll the required boundary conditions do not depend on C, the analytic expression for the integral (8b) reads: here E kin is the sum of the translational kinetic energies of the colliding particles before (θ v ) and after (θ u ) the interaction.
In [17] an extra assumption is made to apply the expression (9) to the realistic Lennard-Jones interaction potential. It is assumed that the transitions mainly take place due to shortrange interaction near the distance of the closest approach. The Lennard-Jones potential in the vicinity of this point is tted into the exponential function. A procedure was elaborated to calculate the parameter a from the viscosity data [17,18].
To get the transition probability as a function of (translational) temperature, equation (9) has to be integrated over Maxwellian velocity distribution of the colliding molecules. In [17] the integral over Maxwellian distribution is calculated (approximately) for co-linear collisions, in the later paper [18] the theory was generalized to realistic 3D collisions. The assumption that the transition is most probable in the vicinity of the distance of closest approach leads to similar mathematical treatment as for the co-linear collisions. Also, the period of rotation is assumed to be large compared to both the period of oscillations and the y-by time. The nal relation for the temperature dependent translational part of transition probability derived in [18] reads (for |∆E| > 0): here T is translational-rotational temperature of the gas in the units of energy. Z (∆E, T) in the present paper corresponds to Z −1 tr in [20,21]. Equation for Z (∆E, T ) in [18][19][20][21] also contains a factor which does not deviate much from 1 and is omitted here. Note also that there is a factor 4 missing in the expressions for transition probability in [17,18]. This mistake which goes back to the very st paper [23] was later recognized and corrected in [19, section 62, p 299].

Modes of oscillations and wave functions
CO 2 is known to be a linear symmetric molecule [4]. It has three modes of vibrations: (1) symmetric stretching; (2) symmetric bending; (3) asymmetric stretching. The bending mode is double degenerate and posses rotational momentum. The whole vibrationally excited state is described by 4 quantum numbers, in Herzberg notation [4]: To nd the wave functions required to calculate the matrix elements (8c) the simplest linear oscillator model is applied. The oscillations are described by independent equations for the stretching modes 1, 3 and the degenerate mode 2: (13) here m are the masses ascribed to the modes (see below), and ω are the measurable oscillation frequencies. It is convenient to transform (13) into polar coordinates x 2 = ρ cos φ, y 2 = ρ sin φ, and look for the solution in the form where the angular part Ψ l (φ) = e ilφ / √ 2π is readily calculated, see e.g. [24, section 27]. Here l is the projection of the rotational angular momentum to the axis of the unperturbed molecule. Equation for the radial part Ψ v l 2 (ρ) then reads: The full wave function is the product of the individual modes: A well known non-harmonic effect which occurs when the energy terms of the order higher than 2 omitted in equations (12), (13) are included is the Fermi resonance. It takes place in CO 2 due to the fact the fundamental frequency ω 1 is almost exactly twice as large as ω 2 . Solutions of higher order obtained by solving the so called secular equations (secular determinant), see [24, section 39] and [26] take a form of the linear combinations of the eigenfunctions (15). In the subsequent discussion the Fermi resonance is not taken into account. Later in section 5.4 it will be shown that as long as the assumptions of the two-modes approach, section 5.1, hold, and the states in Fermi resonance are orthogonal to each other it is not required to explicitly take this effect into account.

Interaction potential as a function of normal coordinates
Speci cally for CO 2 the problem of expressing the interaction potential as a function of normal coordinates (z 1 , z 3 , x 2 , y 2 ) was solved by Herzfeld in [20]. The geometry of the collision is sketched in gure 2. An interaction between structureless atom M and the atom A of the molecule closest to M is considered, in uence of other atoms of the molecule is neglected.x,ỹ,z are the displacements of the atom A from its equilibrium position in the Cartesian system with zero placed in the position of M. Under assumption of small displacementsx,ỹ,z ≪ r the instant distance between A and M is:r The coordinates x, y, z are de ned in another coordinate system with axis z directed along the unperturbed axis of the molecule, and the origin placed into the unperturbed position of A. An exponential interaction potential is transformed as: This is the factorization assumed in (7). Following [20] the geometrical displacements are written as linear combinations of the normal coordinates: The procedure of nding the coef cients α elaborated in [20] is based on the requirement that the momentum and energy balance is ful lled for each individual mode separately-this follows from the orthogonality of non-degenerate normal vibrations, see [4, section 2.11]. The procedure assumes free molecular oscillations-that is, the in uence of the interaction with M is neglected. Assignment of the mode masses in (12) can be done arbitrary. In the nal results only ratios α 2 /m appear, and α 2 depends on the chosen m so that the mass cancels out. In the present work the assignment made in [20] is used. Although α for O and C atoms are substantially different, Herzfeld [20] suggests for modes 1 and 3 to always use 1,3 . For mode 2 an arithmetic mean of α (O) 2 and α (C) 2 is suggested. The masses m and corresponding coef cient α from [20] are reproduced in table 1 below. Substituting (18) into (17) yields: The linearization is justi ed by smallness of the displacements, α 1 z 1 , α 3 z 3 , α 2 y 2 ≪ a −1 .  (26).

Matrix elements
To nd the oscillatory part of the transition probability equations (15) and (19) are substituted into (8c). By doing so the integral in (8c) is separated into three independent integrals for each mode of oscillations. For a transition from CO 2 v 1 , v l 2 , v 3 to CO 2 u 1 , u m 2 , u 3 this integral can be then written as: Since the interaction takes place mainly in the vicinity of the closest approach and the rotation is suf ciently slow it is assumed that the angle θ is not changed during the interaction, and can be taken out of the integral. Expressions for the non-zero matrix elements v 1,3 |z 1,3 |u 1,3 of the coordinate of a linear oscillator are readily available, see e.g. [24] section 23: Calculation of the matrix element of a double degenerate oscillator was addressed in [27]. The integral is transformed into polar coordinates: The rst term of the sum equals to: The analytic solution of the Schrödinger equation (14) is expressed in terms of the associate Laguerre polynomials (see [27, section 5]), and the integration over ρ above follows from the properties of those polynomials and from the normalization. In order to integrate the second term the y 2 is replaced by . This yields nally: The angular momentum l is always changed by ±1 when the quantum number v 2 is changed. The non-zero matrix elements of the double-degenerate linear oscillator found in [27] read: Transformation of the original notation of [27] into equations (24)

Probabilities of the elementary vibrational transitions in CO 2 : a summary
Combining (20), (21), (23), (24) the resulting squares of the matrix elements of the individual modes can be expressed as follows: where: and X are the normalized matrix elements of linear oscillators: In equation (26) with numerical factor: ω i is in eV, a is in Å −1 , i are taken from [20] (same m i and α 2 i were used in [21]), ω i are from [28]. Numerical values of the factors α 2 m i are calculated assuming the carbon atom mass m C = 12 and the oxygen atom mass m O = 16, m CO 2 = 2m O + m C The total probability P of a transition from CO 2 v 1 , v l 2 , v 3 to CO 2 u 1 , u m 2 , u 3 is obtained by combining the oscillatory and translatory parts, see (8a), and averaging over the angle θ: here E are vibrational energies of the corresponding states, A and X are the products of A i , table 1, and the normalized matrix elements (27) of the modes which are changed in the transition. S is a 'steric' factor obtained by averaging the product of the angular terms of (25) over θ. Factors S for all transitions considered in the present work are given in table 2. Following [20] the calculation of S is done on assumption of a homogeneous distribution of θ, the distribution function is f (θ) dθ = 1 4 |sin θ| dθ. The function Z |∆E| , T de ned by (10) can only be used when σ > 1. Instead of this formula in practical calculations, see e.g. [5], a t proposed in [32] is applied: This equation matches (10) when σ → ∞, and gives a correct asymptotic value for |∆E| = 0 as well, see gure 3. For σ it is convenient to re-write equation (11) in a more vivid form: (31) Parameter E 300 σ calculated for different collision partners which may occur in CO 2 plasma is given in table 3. Parameter a is calculated with the formula proposed in [17] as a = 2 · 18.5/ r A 0 + r B 0 , where r A 0 , r B 0 are parameters of the Lennard-Jones potential of the colliding molecules. Values of r 0 applied here are same as in [5]: r CO 2 0 = 3.94 Å, r CO 0 = 3.69 Å,  Generalization of the theory to the case where both colliding particles change their vibrational states appears to be straightforward. In order to do that the oscillations of the atom M have to be added to (16) in addition to oscillations of A. Linearization will then lead to the expressionr = r +z +z M , wherez M is the displacement of M from the equilibrium position in its molecule. Subsequently, (17) receives one more multiplier e az M , and the transition probability (8a) will be multiplied by another term (8c) calculated for the molecule to which the atom M belongs.
The nal result will be expressed by the same equation (28), only factors X and A have to include multipliers related to the elementary quantum jumps in the second molecule. If the directions of the colliding molecules can be taken as completely independent, then their steric factors S are calculated independently as well, and multiplied with each other. ∆E in (28) and (30) is actually the difference of translational kinetic energies before and after the collision. Therefore, in the general case it must include the total change of vibrational energy in both molecules. That is, for inelastic collision of CO 2 molecules of the kind: (29) must be replaced by:

Combined states
In CO 2 the fundamental frequency ω 1 is very close to 2ω 2 , see table 1. Therefore, neglecting terms of higher order over v 1 , v 2 , v 3 and l, the vibrational energy E v of a CO 2 molecule can be approximately written as: v s describes both symmetric modes and is called 'the symmetric quantum number'. The most basic assumption of the two-modes approach to describing the vibrational kinetics of CO 2 is that vibrational energy depends only on v s , v a . It is not critical that exactly expression (33) must hold, but the critical assumption on which all analytic derivations are based is that states with (v 1 , v l 2 , v 3 ) which have same v s and v a all have exactly same vibrational energy. Such states can be gathered into one 'combined state' [v s , v a ]. Subsequently, (v 1 , v l 2 , v 3 ) which belong to [v s , v a ] are called the 'sub-states' of this combined state. Further, very fast equilibration of the sub-states populations is assumed which, together with the assumption of equal energy, implies that the populations of the sub-states are equal as well. In practical calculations as the energy of a combined state E v s ,v a the average vibrational energy of all its sub-states is taken. Those latter are calculated applying expressions more accurate than (33), see [28][29][30][31], which include terms of higher order.
From (34) one can see that if the symmetric quantum number is even, v s = 2n, then v 2 = 2 (n − v 1 ). Condition v 2 0 is ful lled by all v 1 0 which run from 0 to n. For the odd symmetric number v s = 2n + 1, v 2 = 2 (n − v 1 ) + 1. The condition v 2 0 is again ful lled by all v 1 from 0 to n. The full set of the combinations (v 1 , v 2 ) which belong to one v s state can be written as follows: As shown in [27] the values which can be assumed by the angular momentum l depend on v 2 : The total number of sub-states in a combined state-the effective degeneracy of the state [v s , v a ]-equals to:

Average probabilities of transitions between combined states
Let us rst consider a general case where the sub-states have different energies E v 1 ,v l 2 ,v 3 , and their populations are distributed according to Boltzmann formula. Relaxation of a combined state in a collision with unchanging particle M: is realized by the elementary transitions between its sub-states: The probability of the transition between combined states is obtained by averaging over the elementary transitions: The probability P is de ned by (28) with ∆E de ned by (29). The sum v 1 ,v l 2 ∈v s is taken over all sub-states of the state [v s , v a ], the sum l→m is over all possible changes of l in the transition: l→m = l→l±1 or l→m = l→l .
By applying the basic two-modes assumption that (38) is simpli ed as follows: N v s is determined by equation (37), X u a v a stands for the normalized matrix element of the asymmetric mode transitions, see (27): is the remaining product of the matrix elements for the transitions between symmetric modes (or 1 if those modes are not changed). For vibrational-vibrational relaxation between the combined states: realized by the elementary transitions where both molecules change their states: → CO 2 γ 1 , γ λ 2 , γ 3 + CO 2 ν 1 , ν µ 2 , ν 3 an equation similar to (38) can be written for the average transition probability: where, extending (28): Analogously to (39), on assumptions of the two-modes approximation: ∆E is de ned by (32), Y γ a v a , Y ν a u a are de ned by (40).

Detailed balance
It can be shown that ful llment of the condition: is suf cient for the detailed balance relations to be valid for the probabilities of direct and reverse transitions between combined states de ned by equations (39) and (44). From equation (42) it is readily seen that X u a v a = X v a u a . Therefore, if (45) holds for the transition probability de ned by (39), then one can write: (29) for ∆E. Or: For the transition probability de ned by (44): Or, making use of equation (32): Equations (46) and (47) which are valid when (45) is fullled are indeed the detailed balance relations between the probabilities of the forward and backward transitions.

Fermi resonance
Wave functions of the states in Fermi resonance Ψ n are linear combinations of the eigenfunctions (15) denoted here as Ψ i (z) (x n is denoted as z): The total number of functions Ψ n equals the number of terms in the sum. The orthogonality condition 2 reads: Substituting (48) yields: That is: The matrix element V b a of the transition between two Fermi states a and b is calculated as: u = u (z 1 , z 3 , ρ, φ) = (1 + aα 1 z 1 cos θ) (1 + aα 3 z 3 cos θ) is the product of matrix elements (20) and (23). Square of the matrix element of the transition a → b: Integration of (51) over angle θ yields: See (28). Let n are all states a which are described by the linear combinations (48) of the same states Ψ i belonging to one combined state. Subsequently, m are all states b in Fermi resonance belonging to another combined state. Statistical factors Y in (39) and (44) written for the transition between those combined states will contain sums of the squares of matrix elements for all possible transitions between states n and m. Making use of (50) those sums can be transformed as follows: This result implies that the sum taken for transitions between vibrational states described by the single eigenfunctions leads to exactly same result as when the orthogonal linear combinations of the eigenfunctions are taken into account explicitly. That is, in frame of the assumptions of the two-modes approximation and as long as the states in Fermi resonance are orthogonal there is no need to take them into account explicitly.

Selection of relevant processes
The assumption that oscillating molecule is represented as a set of linear oscillators limits the allowed elementary transitions to one-quantum jumps only. Nevertheless, the total number of all logically possible transitions is very large. At the same time, a limited number of the most relevant processes can be de ned based on the simple considerations which will be presented below. This pre-selected set of transitions is shown in table 4. In this table the column 'combined' denotes the end state of the molecules after the collision written it terms of combined states, and the 'sub-state' are the corresponding sub-states after the collision. The initial states are assumed to be: and respectively, where M is a particle which is not changed in the collision. Parameter (SA) in the last column is calculated assuming a −1 = 0.2 Å-last row of table 1. This coef cient, see (39) and (44), helps to sort out the processes whose probabilities can be expected to be negligibly small. Another parameter which helps to reject the likely irrelevant processes is |∆E| since the transition probability is rapidly decreased with increased |∆E|, see gure 3. Let us rst consider collisions of the kind CO 2 [v s , v a ] + M where only one particle is changed. Processes (1). . . (4) in table 4 are all possible transitions of that kind where only one of the two modes-v s or v a -is modi ed. Those processes represent vibrational-translational (VT) energy exchange. Energy exchange between the modes v s and v a takes place in processes (5). . . (7). Transitions which involve quantum jumps of the same modes as (5). . . (7), but with both symmetric and asymmetric quanta changed in the same direction would represent another VT-channel. It can be immediately seen that they can be neglected compared to the processes (1). . . (4). The (SA) of those latter is larger, and their |∆E| is smaller.
In processes (8). . . (11) which describe vibrational-vibrational (VV) exchange only symmetric or only asymmetric mode is changed. The same transitions where the mode numbers are changed in the same direction would be VT-processes competing with (1). . . (4). Applying the same argument as in the previous paragraph leads to the same conclusion that this VT-channel can be neglected compared to (1). . . (4).
Collisions where both particles can change the symmetric and asymmetric modes simultaneously lead either to VT or VV-transfer or to the inter-mode exchange of vibrational energy. That is, only those of them shall be taken into account which can compete with processes (1). . . (11). Same as above, VT-processes (1). . . (4) and VV-processes (8). . . (11) are always faster due to both larger (SA) and smaller |∆E|. Hence, only inter-mode transitions which can compete with (5). . . (7) should be left-those are processes (12). . . (16).
The list of pre-selected processes, table 4, can be further re ned. First of all, the alternative elementary pathways of the processes (2), (5), (9) marked with * can be immediately neglected because their (SA) is orders of magnitude smaller, and |∆E| is exactly same. Second, among the VT-processes (1). . . (4) only the process (2) with smallest |∆E| must be taken into account, since (SA) of the processes (1), (3), (4) is same or smaller than that of (2). Due to the same reason process (5) can be neglected compared to (6). At the same time, (7) cannot be sorted out on the basis of purely qualitative considerations. Although for this process (SA) is much smaller than that of (6) its |∆E| is also smaller.
The outcome of the selection procedure is presented in the rst column of table 4 where only the processes selected for further consideration are numbered.

Transition probabilities
In table 5 all relevant processes picked up from table 4 are listed, together with statistical factors (40) required for the calculation of their probabilities. There are in total 15 processes (including direct and reverse transitions), but only 6 statistical factors Y listed in table 6 are required. To calculate the nominators of (40) equation (41) is reduced using de nitions (35): The quantities X u 1 ,v 2 v 1 ,v 2 are given in the last column of table 6. For transitions where v 2 is not changed  3. It has been shown that the derived relations fulll the detailed balance, and that the Fermi resonance must not be taken into account explicitly within the assumptions of the model.
Inaccuracies which may be introduced in the model by the harmonic oscillator approximation have been already brie y discussed in the introduction. The assumption that vibrational energy is strictly a function of only the two quantum numbers v s and v a which led to the simple algebraic expressions (39), (44) apparently introduces a further inaccuracy. This fact on its own does not discard the derived expressions and does not preclude their use. The theoretical formulas are commonly applied not to nd the absolute values of the transition probabilities, but as scaling laws to nd the rates of transitions for which no experimental data are available. See [5,21]. That is, the nal rate coef cients always contain uncertainties related to the experimental data, and those uncertainties can be large. The review paper [34] suggests typically a factor of 3 and even larger differences for the rate coef cients taken from different sources. The extra error introduced when proceeding from equations (38), (43) to (39), (44) is not of practical relevance if it is below the inaccuracies of the underlying experimental data. To estimate this error equations (39), (44) can be compared directly with the results obtained by applying (38), (43).
There is a dif culty which makes this estimate not straightforward. Namely the lack of an accurate model for vibrational energy which would be valid at large quantum numbers. Several expressions for the CO 2 vibrational energy can be found in the literature. The equation from [28] which keeps terms up to second order in quantum numbers v 1 , v 2 , v 3 , l was used in the present work. This expression was obtained by tting the spectroscopic data for v 1 up to 3, v 2 up to 7 and v 3 up to 5. That is, applying this expression to v s signi cantly larger than 7 is already a wide extrapolation. In the more recent paper [29] an expression with terms up to third order is proposed, states with v s up to 9 were used for the t. This expression which is thought to be more accurate than [28] produces much stronger scattering of the energies of sub-states belonging to one combined state for high v s . This scattering can very likely be an artifact of the polynomial extrapolation of higher order. Furthermore, the effect mostly comes from the terms proportional to l 2 . In a later paper [30] the data set of [29] was re ned, and it was shown that a good t can be obtained even with those terms set to zero. Extrapolation to large v s with the equation from [30] leads to much smaller scattering than both [28,29]. That is, the basic assumption of the two-modes approximation would be ful lled with much higher precision. Expression for vibrational energy from [31] which was used in [15] does not have the l 2 -dependence as well.
It should be also reminded about two further known issues with the SSH theory itself. The rst is that when the SSH expressions are formally applied to high excited states the resulting probabilities may become larger than one. This is a genuine problem of the method of perturbations (Zener's method). This limitation can manifest itself especially at high temperatures. The second issue is related to the assumption of the dominance of short-range interactions. This assumption is known to be incorrect for vibrational-vibrational exchange between asymmetric vibrations with small |∆E| [35,36]. If dipole-dipole interactions do not affect the transitions between symmetric states (the CO 2 is a symmetric molecule with zero dipole moment in ground state), than this latter problem does not affect the derivations of the present paper. It only affects the process 4 in table 5 where a term which takes into account the long-range interaction has to be added, see e.g. [37].

Appendix A. Translation of the results from [27]
Formulas for the matrix elements of a double degenerate oscillator applied here were taken from the review paper [27]. According to table 4 there (ρ Sh is variable ρ in [27]): Sh (v + 1) |l|+1 To reconstruct the absolute values of the matrix elements implied in [27] it is noticed that ρ 1 2 Sh is connected to ρ in the present paper as: