Some Aspects of the Theory of Vibrational Transition Probabilities in Molecular Collisions BY B. WIDOM Dept. of Chemistry, Cornell University, Ithaca, New York, U.S.A. Received 27th November, 1961 An exponential formula commonly used in the discussion of vibrational transition probabilities in molecular collisions is analyzed with the object of determining the extent to which it is dependent on specific (but questionable) assumptions about the form of the interaction energy. Using a principle of Landau, a very general expression is derived for the exponential appropriate to the one-dimensional case. It is shown that the leading term is very insensitive to the precise nature of the interaction, so that this term in its commonly used form has a great degree of generality. The correction terms in the exponential, on the other hand, depend sensitively on the assumed form of the interaction in the region of strong repulsion, so the corrections in current use must be considered very uncertain.1. INTRODUCTION It has in recent years become common practice to discuss vibrational transition probabilities in molecular collisions with the use of the formula, where P is the probability per unit time that the contemplated transition (a de-excita- tion) will occur ; p is the reduced mass of the collision partners ; A(> 0) is the mag- nitude of the change in the oscillator’s energy due to the transition ; B( >O) is the depth of the potential energy minimum in the interaction of the collision partners; II and kT have their usual meaning; and a is the range parameter which occurs when the interaction energy w as a function of distance x between centres of mass of the collision partners is represented by an exponential of the form w = A exp (- x/a) in the region of strong repulsion. The prz-exponential factor in P has been omitted from eqn.(1 . l ) because it is not involved in the present discussion. To obtain the form of the exponential in eqn. (1.1) appropriate to an excitation of the oscillator, rather than to a de-excitation, one has merely to replace A by - A . The first term in the exponential, as quoted in eqn. (l.l), is due to Landau and Teller,l while the remaining two terms, which are considerably smaller in magnitude than the Landau- Teller term, though still important, are due to Schwartz, Slawsky and Herzfeld,s 3 and will be referred to as the correction terms.Critical comparisons of eqn. (1.1) with experimental results have been made by Dickens and Ripamonti4 and by McCoubrey, Milward and Ubbelohde.5 The present discussion is concerned with the validity and degree of generality of eqn. (1.1). The Landau-Teller term and the correction term A/2kT were origin- ally derived 1 9 2 for one-dimensional systems with the explicit assumption of an exponential repulsion between the collision partners and, indeed, the former term still contains the range parameter characteristic of this potential. The correction term D/kT arose3 from a physical argument in no way dependent on an explicit choice of interaction potential, but the argument, though suggestive, is not obviously valid.37 P- exp [ - 3 ( 7 ~ ~ p A ~ a ~ / 2 h ~ k T ) ~ + A / 2 k T + DlkT], (1.1)38 VIBRATIONAL TRANSITION PROBABILITIES The analysis will be restricted to one-dimensional systems, but no special form for the interaction energy will be assumed until it is clear what the effects of such specialization are. The relative translational motion of the collision partners is essentially classical, so in the exponential expression for P the exponent, which arises from the translational motion, can be written as a sum of terms each classified according to its order of magnitude in A, and those terms which vanish as Pr-0 can be discarded. In this way there arise terms, such as the Landau-Teller term in eqn. (l.l), which arb large when ki is small, and also terms, such as A/2kT or D/kT, which are independent of A.By formally treating the energy A as an entity inde- pendent of fi, while letting h+O, the quantum nature of the vibrational motion and the essentially classical nature of the translational motion are both correctly taken into account. For a purely repulsive exponential interaction energy, the Landau-Teller term and the Schwartz-Slawsky-Herzfeld term A/2kT are exact, there being no other terms that survive when E+O. It will be seen that a slightly generalized form of the Landau-Teller term is still the correct leading term for a very general class of potentials. The correction terms, however, will be seen to be determined in general by certain detailed features of the interaction potential in the region of strong re- pulsion, and as these details are as yet unknown, both experimentally and theoret- ically, for real interactions, the correction terms must be considered to be in great doubt.The uncertainty is not over whether the Schwartz-Slawsky-Herzfeld terms A/2kT and D/kT should be included in eqn. (l.l), because it will be established that the former is surely, and the latter is probably, a correct %-independent term for general potentials ; but it will appear that these are by no means the only possible terms of this order of magnitude, and also that terms intermediate in magnitude between these and the leading (Landau-Teller) term may also be present. 2. TRANSITION PROBABILITY PER COLLISION Imagine an oscillator and an impinging particle in interaction in a one-dimensional system.Let be the displacement of the oscillator from its equilibrium position and let x be the distance between the centres of mass of the incident particle and of the oscillator (or of the molecule which contains the oscillator). The interaction energy W(x, c) between the two molecules, in the cases of interest here, may for all practical purposes be assumed linear in the oscillator co-ordinate.1 Thus, In any relevant initial or final state of an oscillator to which the theory might reason- ably be applied, the expectation value of is negligibly small, and it will be assumed here that it vanishes : The potential w(x) is then a typical intermolecular interaction energy, determining the relative translational motion of the two colliding particles both in the initial and final state of the oscillator.The forcef(x) perturbing the oscillator is not so familiar a quantity as is w(x). If W(x, r) were, say, a function of x-C alone (as is often as- sumed), then f ( x ) would be simply w'(x). More generally, however, W must be expected to be the sum of two functions, one of x and one of x-5 (imagine the oscillator to be contained within a large molecule), so that in reality w(x) and&) are quite independent. Alternatively, one can describe the situation by saying that the force which determines the relative translational motion, - w'(x), does not consist only of the reaction, -f(x), of the force on the oscillator, but also of an <5> = 0. (2.2)B. WIDOM 39 additional force which would act between the two colliding particles even if no force acted on the oscillator itself.However, while not much is known about f ( x ) , its effects appear only in the pre-exponential factor in eqn. (l.l), and since the present discussion is restricted to the exponential factor alone, the nature of f(x) is irrelevant. That the magnitude of the transition probability per unit time depends almost entirely on the potential which determines the relative translational motion, and hardly at all on the perturbing force, is the most interesting point in the theory. The probability p that the oscillator will undergo the transition i-+j due to collision with the incident particle may be calculated reliably by the method of distorted waves.6 Alternatively, it may be calculated by the method of perturbed stationary states,6 with the additional assumption that the perturbation of the oscillator energy levels and wave functions by -f(x)c is adequately given by first-order perturbation theory. In either case, it follows from eqn.(2.1) and (2.2) that the probability p is given by the following formula, essentially due to Zener : 7 8pt2. - rJ f (x)G(E, x)G(E +A, x)dxI2, = h2 JE(E +A) - co where ( i j is the matrix element of 5 taken between the unperturbed initial and final states of the oscillator, where E is the smaller of the initial and final relative trans- lational energies in the collision (so that E+ A is the larger), and where for any positive energy E the wave function G(E, x) is the well-behaved solution of r2E+E-w(x) 2p ax2 1 G(E, x) = 0, which is normalized so that as ~ 4 ~ x 3 , where w(x) vanishes, G(E, X) - cos (J2pExlft + 6) (2.9 with 6 a real number which depends on E, and on the form of w(x), but which is independent of x.Either of two forms of the intermolecular potential w(x) might reasonably be envisioned for this one-dimensional system. (i) The interaction becomes infinitely repulsive at a finite value of x (which, without loss of generality, can be taken to be x = 0), as would be the case if w(x) were proportional to a negative power of x. (ii) The interaction is finite for all finite x, but becomes infinitely re- pulsive as x-+ - co, as would be the case if w(x) were of the form A exp (- xla). In case (i), the wave functions G in the integral of eqn. (2.3) are to be taken identically 0 for x<O (so that the integration can be restricted to the interval 0, co) and the well-behaved solution of eqn.(2.4) for x>O is that which vanishes at x = 0. In case (ii) the well-behaved solution of eqn. (2.4) is that which vanishes as x-+ -a. The crucial step in the discussion is now the recognition of the important general principle of Landau,s according to which, if the translational motion is essentially classical, then the integral in eqn. (2.3) is f(x)G(E, x)G(E + A, x)dx - exp JW -(E +A)x’(w)dw- where x(w) is the function inverse to w(x), and x’(w) is its derivative. The pre- exponential factor, containing A only multiplicatively, is not relevant here and has40 VIBRATIONAL TRANSITION PROBABILITIES been omitted. Eqn. (2.6) represents the asymptotic evaluation of the integral on the left-hand side, as A+O. It is here that one sees that the exponential depends only on the potential w(x) and not at all on the forcef(x).The power and generality of Landau’s principle, in the form of eqn. (2.6), can be tested by applying it to three special cases for which the integral on the left-hand- side is known exactly : (9 w(x) = A exp (- x/a), f(x) = B exp ( - x/b). This is the exponential interaction; in the special case b = a the exact result is due to Jackson and Mott 9 and has been the basis of most previous discussions of vibrational transition probabilities. There is no difficulty in extending the exact result to general b, and as R+O the integral on the left-hand-side of eqn. (2.6) becomes where the pre-exponential factor contains k only multiplicatively and has been omitted. This is precisely the exponential given by the right-hand-side of eqn.(2.6). That the exponential depends on a but not on b is due to the fact that it depends on w but not onf. (ii) w(x) = D[exp (-x/a)-2 exp (-x/2a)], f(x) = Aw‘(x) ’-“his is the Morse potential, for which the exact evaluation of the left-hand-side of eqn. (2.6) is due to Devonshire.10 Letting 5-0 and omitting the pre-exponential factor, Devonshire’s result reduces precisely to that calculated from Landau’s principle, viz. , where for any positive E, +(E) = JETD arc tan J T D - 3 In (1 + E/D). (iii) W ( X ) = A/X2, f(x) = B/x2. This is the inverse square potential; the exact value of the integral 11 yields for its exponential factor, in the limit A+O, (1 + A/E)- dPA/2E2 again in agreement with Landau’s principle.Thus, whether w(x) becomes infinitely repulsive at x = --GO or at a finite value of x, and whether it is purely repulsive or also partly attractive, eqn. (2.6) gives the correct result. The transition probability per collision, which is essentially the square of the quantity in eqn. (2.6), may, therefore, be considered known explicitly for any intermolecular potential w(x). If in eqn. (2.6) one makes the assumption that A<E, and so discards all but the leading term in A, the result is identical to what would be obtained from the so-called semi-classical theory 12 if in the latter theory the collision integral were evaluated in the asymptotic limit li+O. This was clearly recognized by Landau and Teller 1 and, indeed, was the basis of their analysis.Thus, the semi-classical theory contains nothing which is not already contained in eqn. (2.6). Furthermore, it is capable of yielding only the leading term, but not the correction terms, in expressions such as that of eqn. (1.1).B. WIDOM 41 3. TRANSITION PROBABILITY PER UNIT TIME The de-excitation probability per unit time P is obtained from the transition probability per collision p(E) by * P = J p ( ~ ) z ( ~ ) d ~ , (3.1) 0 where z(E)dE is the number of collisions per unit time suffered by each oscillator, in which the initial relative translational energy is between E and E+dE. The integral is then to be evaluated in the asymptotic limit fi+O. The exponential part of p(E), as given by eqn. (2.6), and the factor exp (- E/kT) in z(E), are the only factors in the integrand which contribute to the exponential part of P, and the result, from eqn. (2,3), (2.6) and (3.1), is that where E* is the solution of co (n !)- 'A"F(")(E*) = B/,fGkT, n = 1 and where F(n)(E) is the nth derivative of the function F(E) defined by F(E)=J&- * X'(W) - E (3.3) (3.4) To evaluate these quantities one must know the detailed behaviour of x'(w) for large w, that is, the detailed behaviour of the interaction potential w(x) in the FIG.1.-The function x(w) is the inverse of the intermolecular potential w(x), and its derivative x'(w) is shown as a function of w. Curve a represents the case of a purely repulsive interaction while curve b, with two branches, represents the case in which the interaction energy has an attractive region of depth D.region of strong repulsion. Such knowledge would lead to asymptotic expressions for F(E) and its derivatives at high energies, and this is what is required since E* is large. Two possible forms of the function x'(w) are shown qualitatively in fig. 1, curve a representing the function when the interaction is purely repulsive, and curve b (which consists of two branches) representing the function when w(x) has an attractive region of depth D.42 VIBRATIONAL TRANSITION PROBABILITIES All that one can say with reasonable assurance about the behaviour of x’(w) for large w is that it is of the form x’(w) = -~aw-~[I+g(w)] (a>0, N>1) (3.5) where For example, the exponential potential w = A exp ( - x / a ) is the special case N = 1, g(w) = 0, while the inverse power potential w = Ax-m is the special case a = m-lAlim, N = 1 + l/m, g(w) E 0.Note that as far as Nis concerned, the exponential potential is an inverse power potential in which the power is infinite. So long as the power is large, whether it be finite or infinite, N in eqn. (3.5) will be close to, and not less than, unity. This observation is important in establishing the generality of the leading term in the exponential in P. It follows from eqn. (3.2) (3.5) that, except for terms in the exponential which vanish as B-+O, g ( 0 ) = 0. P - exp [ - E*/kT + (,/aE)AF(E*) + A/2kT] with the first two terms themselves given by I‘(N++) J2TaAkT 1/(NS3) h 1 - E*JkT + (J&i/ft)AF(E*) = - + correction terms (3.7) The leading term in the exponent is that given explicitly on the right-hand-side of eqn. (3.7).It is the generalization of the Landau-Teller term to arbitrary potentials, and reduces to it for the exponential potential. Since, as noted above, N is close to 1 for all strong repulsions, whether exponential or not, the leading term in eqn. (1.1) remains nearly correct for all realistic cases. The situation as regards the correction terms is quite different. The Schwartz- Siawsky-Herzfeld term A/2kT appears explicitly in eqn. (3.6) as a correct A-inde- pendent term for the most general potential, but additional terms of the same (and, perhaps, even greater) magnitude are contained among the correction terms on the right-hand-side of eqn.(3.7), and these terms require for their evaluation a detailed knowledge of the function g(w) in eqn. (3.9, that is, a detailed knowledge of the inter- action potential at high energies. Foi example, if g(w)-b/w for large w, then the complete exponent in P consists of the leading term plus the correction terms A/2kT-b/NkT, so that in this case there is an %independent correction in addition to A/2kT. There is, of course, no Teason whatever to assume that g(w) behaves like l/w for large w ; for a Morse potential, or for a Lennard-Jones 212-n potential, g(w) behaves like l/w* for large w, and then correction terms appear which, as far as their order of magnitude in li is concerned, and probably their numerical mag- nitude as well, are intermediate between the leading term and the additional fi- independent corrections. The correction term D/kT in eqn.(1.1) might appear in the following way. Suppose the lower branch of curve b in fig. 1 is represented by xf(w) = -a(w+ D>-N, which is qualitatively correct. Then g(w)- -ND/w for large w, s0 from the dis- cussion of the preceding paragraph it follows that the correction terms are precisely A/2kT+D/kT, and no others, just as in eqn. (1.1). This argument makes it highly likely that D/kT is also a correct h-independent term in the general case, but all the ditficulties concerning the probable presence of other terms remain.B . WIDOM 43 The conclusion one must draw from the foregoing analysis is that the leading term in the exponent of P is known in great generality, but that the corrections are still very uncertain. This research was supported by the U.S. Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command. 1 Landau and Teller, Physik. 2. Sowjet., 1936, 10, 34. 2 Schwartz, Slawsky and Herzfeld, J. Chem. Physics, 1952, 20, 1591. 3 Schwartz and Herzfeld, J. Chem. Physics, 1954, 22, 767. 4 Dickens and Ripamonti, Trans. Faraday Soc., 1961, 57, 735. 5 McCoubrey, Milward and Ubbelohde, Trans. Faruday SOC., 1961, 57, 1472. 6 Mott and Massey, The Theory of Atomic Collisions (Oxford University Press, Oxford, 1949), 7 Zener, Physic. Rev., 1931, 37, 556. 8 Landau and Lifshitz, Quantum Mechanics (Pergamon Press, London, 1958), pp. 178-183. 9 Jackson and Mott, Proc. Roy. SOC. A, 1932,137,703. 10 Devonshire, Proc. Roy. SOC. A, 1937, 158,269. 11 Widom and Bauer, J. Chem. Physics, 1953, 21, 1670. 12 Zener, Physic. Rev., 1931, 38, 277. 2nd ed., chap. 8.