Resonance and Non-Resonance Intermolecular Energy Exchange in Molecular Collisions BY E. E. NIKITIN Institute of Chemical Physics, Vorobyevskoye chaussie 2-b, MOSCOW V-334, U.S.S.R. Received 17th January, 1962 Two approximations involved in the calculation of the probabilities of transition between mole- cular vibrational or electronic terms are discussed. An exact solution for two terms is found. In the constant velocity approximation, when the effect of the turning point is neglected, this solution may be used in calculating the charge exchange cross-section in accidental resonance or intramolecular energy exchange in the collision-induced Fermi resonance in three-atomic molecules. The prob- ability of transition between non-parallel vibrational terms is calculated in the semi-classical ap- proximation, the effect of the turning point being taken into account.This is of importance for inelastic molecular collisions involving two or more vibrational modes. The general connection between the Landau-Zener and Landau-Teller approximations is considered and the validity of the classical relative motion of molecules is discussed. Modern theories of inelastic atomic and molecular collisions resulting in electronic or vibrational excitation are based largely on calculations of the probability of non- adiabatic transition between electronic or vibrational terms. In many cases these processes induced by non-adiabatic interaction of intra- and intermolecular motions, can be described in terms of the semi-classical approximation of relative translational motion even for the transition close to the turning point.1 But the more simple cases are those when the main contribution to the transition probability is from the regions that are far enough from the turning point.One example is the crossing of zero-order terms far from the turning point; then the non-adiabatic transition probability can be calculated according to the Landau-Zener formula 2, 3 corrected for the high energy limit by Bates4 and Mordvinov and Firsov.5 In deriving the formula it is assumed that the perturbation matrix element e2 is time independent and the ratio of 82 to the term splitting at infinity is diminishingly small. However, in many cases the asymptotic splitting is not sufficiently large, as in the accidental resonance-type reaction discussed by Bates.6 On the other hand, vibrational terms in non-elastic molecular collisions run almost parallel for all intermolecular distances, and the corresponding transi- tion probability is given by the Landau-Teller formula7 derived for the three- dimensional system by Schwartz and Herzfeld.8 The cross-sections for non- elastic molecular collisions differ very markedly in these cases.In particular, crossing of electronic-vibrational terms in collisions of two NO molecules results in a con- siderable increase (by 3-4 orders) of the probability of vibrational transition from the ground to the first excited vibrational level, as compared to the respective transi- tion probability for 0 2 or N2.9. 10 In this connection it is of interest to find an exact solution of a non-stationary two-state wave equation that might be used for inter- preting all the intermediate cases-from crossing to parallel running terms.Further, we shall consider non-adiabatic transitions in the Fermi resonance induced by molecular collisions. 14E. E. NIKITIN 15 Such processes that are due to molecular collisions are very important for spectro- scopy and kinetics. Besides, solution of this problem not only gives the transition probability for all closely spaced vibrational terms in complex molecules, but also determines the zero-order wave function that can be used in calculating the vibra- tional relaxation time in gases composed of polyatomic molecules. The Fermi-resonance type interaction in complex molecules is known to be one of the main effects that give rise to strong interaction of normal modes.Under stationary conditions the interaction is manifested in abnormal shifts of vibrational terms that are in resonance, and in the mixing of corresponding wave functions. Under non-stationary conditions, when vibrational terms approach each other adiabatically becoming resonant under the influence of external force, the induced Fermi resonance is expected to bring about considerable intramolecular intermode energy exchange. Let us consider a linear three-atomic molecule in its completely symmetrical vibrational states. Now, we suppose that the energy difference 81 between the stretching mode quantum hcol and the doubled bending mode quantum 2ha2 is small as compared with ha1 (here and hereafter h is the Planck constant divided by 2n).In this case the anharmonicity mixes to a considerable amount only functions of the two zero-order states corresponding to energies of hwl and 2ho2; let these functions be 41 and 4 2 . When an external force action is adiabatic with respect to transition between all zero-order vibrational levels, we may restrict ourselves to two function bases. When TO is the characteristic time of force action these conditions may be written as follows : For molecular collisions TO may be taken as p/u, where p is the range of action of the intermolecular potential, and 0 is the relative velocity of molecules. &2/h, 1/20 <mi. 1. THE CONSTANT VELOCITY APPROXIMATION The non-adiabatic wave function may be written as Y = al+l exp ( -$E1dt)+a,42 exp ( - i P 2 d t ) , where El and E2 are zero-order energies without anharmonicity; equations hia, = E~ exp [kbEdf]a2, hia, = E~ exp - - AEdt a,, AE = [ b s 1 When the minimum adiabatic splitting of exact terms (i.e., exact with respect to two function bases) is much smaller than the asymptotic zero-order term splitting, we may put E = y t and ~2 = const.Solution of (3) under these conditions leads to the Landau-Zener formula. However, in our case at least on one side of the crossing, the splitting A23 is not great as compared with ~2 and the Landau-Zener approximation is not valid. We assume that the condition A E % E ~ is fulfilled near the turning point of relative motion of colliding molecules, and approximate the splitting AE by the function AE(t) = ~~a exp [ - F o ~ / E , ] + E,. (4) This corresponds qualitatively to an exponentially decaying intermolecular ex- change interaction and contains two essential parameters-the term splitting at infinity ~1 and the force difference AF at the crossing point.The value a must not16 INTERMOLECULAR ENERGY EXCHANGE enter the final expression for transition probability as long as it is connected with the time reference scale. Here we put v to be a constant because the turning point is far from the crossing. As to the time dependence of Q, it can safely be neglected. This is due to the fact that the change in the anharmonic constant in molecular collisions is small. Introducing new variables z = ~ 2 t / h , a = AFvh/~1~2, p = 4 ~ 2 , (3) can be written in the form s da12 - 1 2 - exp [+i f(~)dz]a,,~; f(z) = a exp (-az)+p, dz where the plus and minus signs in the exponent correspond to the first and second subscripts in a.After replacing c12 = a12 exp [&(i/2)Jf(z)dz] this system reduces to one equation, One of its solutions is cl(z) = A exp (-z/2)z'"*~(is1, 2is0 + 1,z) (7) where z = - (ai/a) exp (- az), s = (1 + p2/4)*/a, s1 = -p/2a+s0, and is the conftuent hypergeometric function. Other solutions of (6) can be ob- tained from (7) by replacing 0 by another linearly independent solution of the cor- responding confluent hypergeometric equations. The asymptotic wave function (2) at z+ + 00 is As follows from (3), 41 and 4 2 are stationary wave functions at z+--co, but at z+ + 00 the stationary wave functions are linear combinations of 41 and 4 2 .Apparently in the limit of small interaction, = 41+m142 correlates with $1 and To calculate the probability of non-adiabatic transition between two stationary states from z = + 00 to z = - 00 (e.g., $I+) we need solve (6) under the boundary condition (8) with p+ = 0. Eqn. (7) is seen to be such a solution. The factor A must be found to satisfy the asymptotic behaviour c1(z) at z+O (z- + GO), $2 = as141 - 4 2 with 4 2 . From the condition of crossing we have a<O and Im(z)>O and thereby On the other hand, the following asymptotic expansion is valid for 1 z 19 1,11 A = exp ($nso)/(l + a2si)+. (10) where s1 +s2 = Bo. Thus, the transition probability ~($1-41) = p11 is given by and from the normalization condition it follows that p($1-)42) = 1 -p11.E.E. NIKITIN 17 To obtain the transition probability of the reverse process (z = - co -+z = + co), This choice we must use in (7) a solution of (8) that would vanish at T = -a. is accounted for by the boundary conditions of the transition, I c , I +O, l j c , I 3 1 at 2 3 - co. (1 3) An appropriate solution is (1 1) exp (z)Y (1 + is2, 1 + 2is0, - z), and the corres- ponding expansion is r( - 2isJ 1x2 is,) r(1- is,) r(l +is, +2is,) Y = + (-z)-2iso, I z I Q 1. From (13) we have Thus, comparing the behaviour of c1 at I z 14 1 in (14) with that of (8) corresponding to we obtain the following expression, Allowing for the identity 1 + a2sf = 2a2s0s17 (12) and (17) may be rewritten in the simple form These equations can be shown to be valid not only for crossing terms (a < 0, p> 0), but for non-crossing zero terms as well.In the latter case we must formally put p c 0 to reverse the terms at z+ + 03. To obtain the transition probability 9 1 2 for double passing of the region of great non-adiabacity we can use the relation, Here, however, 9’12 is the transition probability averaged over a small range of velocity changes. After such an averaging, all the terms that are due to interference between a1 and a2 in the second passage vanish. Thus, from (18) and (19) we obtain 9 1 2 = 2P12P22. (19) sinh2 (2ns0) This expression takes the simplest forms under the following conditions. Then, and (20) becomes the Landau-Zener formula, (i) Great asymptotic term splitting and small velocity : &1/&2% 1, 2n$/AFhv% 1.2ns, -+2x~z/AFhv; s2, so-+ CQ P12 = 2 exp (-2n~;/AFhv)[l- exp (-2n$/AFhv)].18 INTERMOLECULAR ENERGY EXCHANGE (ii) Exact resonance to the zero approximation: EI = 0. Then, introducing for convenience a new parameter-the effective range of action of intermolecular forces p = &l/AF-we obtain 9'' = +~h-' (m2p/hv). This result coincides with the formula devised by Rosen and Zener,l2 if the oscil- lating factor in the latter is replaced by 3. (iii) The great velocity of molecules : 274, 27rne:-g AFhv. After expansion (20) becomes The maximum value of 9 1 2 in all cases is 9 1 2 = 3. 2. THE CLASSICAL MOLECULAR MOTION APPROXIMATION In the above section we assumed the condition that the turning point is suf- ficiently far from the region making the main contribution to the non-adiabatic transition probability.This is valid if the velocity change Av during the time of strong interaction At is small as compared to u. Thus, e.g., for crossing terms we may write where m is the effective mass of colliding molecules. Then the above-mentioned condition will be A P v ( A ~ ) ~ / ~ ~ - 1 and Av - (F/m)At, (24) mv E()f<l. AFv In thermal molecular collisions this condition is in many cases equivalent to the low value of intermolecular energy at the crossing point (or at the region of essential term divergence if the terms do not cross) to the mean thermal energy kT. How- ever, in cases when the change in splitting A E is relatively small and its asymptotic value is sufficiently large, the main contribution to the transition probability can also be that from the region near the turning point.Then the constant velocity approximation is not valid and all the equations of $ 1 fail at the limit a+O. But this case is of practical importance because it takes place in molecular collisions resulting in translational-vibrational energy exchange involving only one vibrational mode. Fortunately, for small a, calculation of 9 1 2 can be carried out by the perturbation method. To follow the transition from constant velocity approximation to the more accurate classical molecular motion approximation that takes into account the velocity change of colliding molecules close to the turning point, let us take as a zero-order basis the functions $1 and $2. Then, splitting AE' of these zero-order terms (in 82 units) and the perturbation matrix element V12 are AE' = 2 sin-' #+a exp(-"') cos #, V12 = $a exp(-"') sin #, (26) where cotan 4 = 3p.When 9 5 2 < 1, the exact expression (20) can be obtained ap- proximately by using the perturbation method. For instance, at j?+ - co(4-2//3), the asymptotic expansion of (20) givesE . E. NIKITIN The same result can be obtained from the equation, 19 P12 = 2 ~ ~ ~ m V , , e x p ( i ~ ’ d r ) d z ~ ’ = The last expression does not depend on a because the integrand is invariant (with an accuracy to the phase factor) when replacing z+(z+const.). But this is not a case for classical motion approximation. Allowing for the effect of the turning point we have to replace function a exp (- az) in (28) by another function $(z) that accounts for the true molecular motion in collision.In our approximation the intermolecular interaction vanishes exponentially with the molecular separation, so that we have X(z)-a exp (- a 1 z 1). On the other hand, at small z that corres- I = I-+a pond to close molecular approach, ~ ( z ) should have a maximum corresponding to the turning point. Thus, the allowing for the turning point leads to the expression, As an example we can take ~ ( z ) = (a/2) cosh-1 az. perturbation method using terms in (26) with p = 0 gives Calculation according to the g12 = (7~a/4.)~ cosh-’ (7~12). gI2 = sin2 (za/4a) cosh-’ (n/a). (30) (31) The exact solution of the corresponding equation found by Rosen and Zener 12 is Thus, it may be seen how (30) and (22) approximate this exact solution for 9’12.For a real case, the function ~ ( z ) must be calculated as a perturbation matrix element, the time dependence of which is determined by the classical motion of elastic-scattered molecules. Taking the most common model used in the theory of vibrational-relaxation of molecules, we assume that the exponential repelling intermolecular potential is of a characteristic length p and the perturbation is linear with respect to the vibra- tional co-ordinate x. Then ~ ( z ) will be ~ ( z ) = (a/4) coshe2 (taz). Calculation according to (29) gives Using the asymptotic expansion and introducing (34) and (33), we obtain, after averaging the rapid oscillating cos2 $(&a), either the Landau-Zener expression (if /?a <O), or (27) (if aa> 0).On the other hand, using (33), we can consider the limiting case of small a, when the last factor may be dropped.20 INTBRMOLECULAR ENERGY EXCHANGE The coefficients a in the exponentials of (25) and (32) and the pre-exponential can be conceived as having a different meaning, as we know beforehand that the transition probability must be proportional to the squared matrix element of per- turbation. For the above-mentioned model we put where x12 is the transition matrix element of the vibrational co-ordinate, AEI the additional term splitting at the turning point with respect to asymptotic splitting 81. The general computation of (33) at any values of parameters /3 and a is not difficult for pa> 0, because in this case @ reduces to the hydrogenic s-wave function for the wave number k = a/l p I and R = ap/2a2, values of which are tabulated.But in many cases (e.g., for vibrational excitation) the parameter p/a is sufficiently large, so that the last factor in (33) can be put in asymptotic form, (36) I is bounded. is large, r = - = a hu a2 h2v2 B E1P -=- Apparently for A E ~ / E ~ Q 1, we can put A E ~ -rnv2/2, so that velocity, and the averaging procedure does not affect this factor. does not depend on 3. AVERAGING OVER THERMAL DISTRIBUTION Averaging of 9 1 2 over the impact parameter for the spherical-symmetrical inter- action is not difficult, in principle, and can be performed for most of the cases using, for instance, the “modified wave-number method” set out by Schwartz and Herzfeld.8 The averaging over velocities is more complicated because on the classical description of relative molecular motion the effect of inelastic transition on this motion is neglected.This means that for non-crossing terms the kinetic energy of molecules at infinity is defined with an uncertainty 81. If the latter is neglected, the averaging gives though the principle of detailed balancing requires When kT, the difference is not essential, because (37) and (38) coincide to the zero order. Then (38) can be fulfilled if 9 1 2 and 9 2 1 are modified to the first order that does not give any appreciable change in probabilities as such. This is a usual pro- cedure for the derivation of a semi-classical relaxation equation,13 but when el + kT all the above functions must be symmetrized before averaging.14 The reason for this is as follows.We know that in exact quantum-mechanical calculation the transition probability must be a symmetrical function of initial VQ and final vf veloc- ities of molecules. In the limiting case of quasi-classical motion we must obtain the above equations, where v is sume symmetrical function of vf and uf. But for any symmetrical function we can write (912) = (912) (37) <912>/<921> = exp (&llkT) (38) v(vi,vf) = virfrsl/2mvi+ . . . (39) where the plus (minus) sign corresponds to deactivation (activation). The first two terms of this expansion with respect to the small parameter sl/mv,2 do not depend on the kind of the function ~ ( u Q Y ~ ) , while the others do. The latter conhe ourE. E. NIKITIN 21 application of classical motion approximation to calculation of transition probab- ilities.The classic motion approximation is valid when the transition probabilities are not appreciably affected by the relative uncertainty in molecular energy or velocity of the order (~/mv:)2. This is the order of terms neglected in (39). The introduction of (39) in (33) gives the transition probability that can be de- rived using the method of distorted waves and it has been used by Schwartz and Herzfeld 8 to obtain a quantum-mechanical version of the Landau-Teller formula. It may be seen that classical motion approximation introduces an uncertainty factor q-exp rr:ip - ( - m::*)2)2] in the Landau-Teller formula. Now v* is velocity that represents the main contribution to the average probability. Averaging over velocities by the steepest descent method gives- = x* = (n2mp2&:/2h2kT) and q-exp [i$/(kT)2x*]. Almost in all cases, q is close to unity and the corresponding correction is usually neglected even in the quantum-mechanical calculations. Upon averaging, (33) can be used for calculations of the vibrational transition probabilities in many-atomic molecules for which the non-parallel vibrational terms are very probably due to participation of different vibrational modes. m(v*)2 2kT The author is greatly indebted to Prof. N. D. Sokolov for helpful discussions. 1 Nikitin, Optika Spektr. Russ., 1961, 12, 452. 2 Landau and Lifshitz, Quantum Mechanics (Gostekhizdat, Moscow, 1948). 3 Zener, Proc. Roy. SOC. A , 1932, 137, 696. 4 Bates, Proc. Roy. SOC. A, 1960, 257, 22. 5 Mordvinov and Firsov, Zhur. Exp. Theor. Phys. Russ., 1960, 39, 437. GBates, Proc. Roy. SOC. A , 1959, 253, 141. 7 Landau and Teller, Physik. 2. Sowiet, 1936, 10, 34. 8 Schwartz and Herzfeld, J. Chem. Physics, 1954, 22, 767. 9 Nikitin, 8th Int. Symp. Combustion (Waverly Press, 1961). 10 Nikitin, Optika Spektr. Russ., 1960, 9, 16. 11 ErdClyi, Magnus, Oberhettinger and Tricomi, Higher Transcendental Functions (McGraw Hill, 12 Rosen and Zener, Physic. Rev., 1932, 40, 502. 13 Hubbard, Rev. Mod. Physics, 1961,33,249. 14 Nikitin, Optika Spektr. RUSS., 1959, 6, 141. 1953), vol. 1.