THE PHYSICAL ORIGIN OF FITTING LAWS FOR ROTATIONAL ENERGY TRANSFER.

An historical overview of the various parameterised forms describing RET processes (fitting laws) is presented. The physical models behind these "laws" are compared. Particular attention is paid to the role of angular momentum constraints and to the energy depen- dence of the state-to-state cross-sections. Finally it is shown how general trends can be inferred from the topology of the intermolecular potential energy surface.


INTRODUCTION
Many of the collision systems in which rotational energy transfer (RET) has been studied are too large for accurate close-coupled quantum calculations. Even though the problem may be somewhat simplified for these heavier systems throuqh the use of the fixed-nuclei or energy sudden (ES) 61 I factorisation , the accurate calculation of the basis cross-sections is still a formidable and often impossible task since the intermolecular potentials, particularly of electronically excited molecules, are not known with any precision.
An alternative approach over recent years has been to look for simple parameterized fitting laws to describe the cross-sections. We describe here below the various forms which have been proposed in historical order.

FITTING LAWS
The earliest of these was the exponential gap law (EGL) 2,3 which was advanced over a decade ago in order to interpret infrared chemiluminescence experiments. The EGL proposes that the efficiency of a particular rotational transfer channel decreases exponentially as the amount of energy exchanged, compared to the average kinetic energy, 4,5 increases. Levine, Bernstein and coworkers later realised that the EGL could be justified from theoretical arguments based on information theory and surprisal analysis. Linear surprisal plots imply kjj, C(E) k(Ojj exp(-E)IEjj I/E) (1) 2 where E I/2 lU + Bj(j+I). C(E) and E) are (possibly energy dependent) empirical parameters cho-(0;," is the prior sen for a best fit to the data. k-.
FITTING LAWS FOR ROTATIONAL ENERGY TRANSFER 63 statistical rate constant. Heller and later Sancturary were able to approximately describe the energy dependence of the EGL using very simple semi-classical mechanics. However while the information theory and thermodynamic approach can be used to justify expressions such as the EGL it is still by no means clear, on the microscopic dynamical level, why rotationally inelastic cross-sections should be so insensitive to the intermolecular potential as to allow such a simple two parameter reduction of RET data sets. Furthermore the EGL only "agrees" with experimental and numerically calculated data sets in a very broad sense, the detailed comportment of the cross-sections particularly at large energy gaps is not well described.
2.2 Power laws 8 Some years ago Pritchard and coworkers presented a slightly modified form of the EGL in which the scattering amplitude varied as the inverse power of the energy gap. This power gap law (PGL) was found to give somewhat better agreement with experimental data for the Na2(AIT +) + Xe 9 system than the EGL. In a later publication they extended their ideas by explicitly including the quantum mechanical factorisation resulting from the sudden approximation to create a combined fitting and scaling law known as the infinite order sudden power law (lOS-P) 64 B. J. WHITAKER AND Ph. BRECHIGNAC kj/j, (2j' + 1) exp,(E. E. )/kT x T. ( (4) where (3 is the in plane angle between the initial and final angular momentum vectors,

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More generally Derouard finds that the multipolar rate constants may be written 2j' +1 rr(2 + 1) PK (cos (3) k0 dJ (6) The connection between the quantum mechanical expressions and these latter result is to be found in the large quantum number limits of the Racah coefficients in the ES-factorisation. For the basis rate constants Derouard chooses ko/j (' ( + 1/2)-" (7) i.e. using the fact that for J >>1, ( , -1 nels. The rotational constant of 1 2 is .029 cm and the rovibronic spectrum is consequently fairly dense. This means that at room temperature the number of energetically accessible channels is typically 200. The data however shows that quantum jumps greater than 40 are extremely unlikely FITTING LAWS FOR ROTATIONAL ENERGY TRANSFER 67 events ; the cross-section for j + 40 is three orders of magnitude smaller than that for j +Z when /.1 of the v(0-16) 1T.+ 3]+ band is exg u cited. Dexheimer and coworkers found that it was impossible to fit these data using a power Law and that it was necessary to postulate an additional ad,-hoc constraint on the angular momentum (and another freely adjustable parameter) .18 The best functional form for the basis set was found to be for collisions with the isotopic pairs 3"4He and H2, D 2 at fixed thermal energy (see Fig. I). They conclude that the parameter * is 9roportional to 1/2 which is consistent with the hypothesis of momentum constraints. Our intuition then suggests that the physical basis for the fitting laws may be found in the conservation of angular momentum rather than in the conservation of energy. Thus the initial ideas of the energy gap laws, such as the EGL, are transposed into an angular momentum language, through the relation E.
Bj(j + 1) in the power laws, such as the IOS-P and ECS-P, and finally angular momentum constraints are explici-, tly included in the parameter (IOS-EP). Although it is rather abusive to call eq. (8) "rotational rainbow model" the understanding of RET has gained very much from these experimental findings.
V,, (R)P,. (cos e) (9) and that the short-range repulsive part of Vz(R) is responsible for the large quantum jumps Aj which we have seen to be of particular interest, since it is in these channels that we observe the strongest deviations from statistical energy redistribution.
The basis of the first condition comes from considering the few systems which cannot be con- Here the collision is adiabatic, indeed there is evidence that some trajectories lead to the formation of a long-lived collision complex, and it is interesting to note that, while the 3 adjustable parameters ECS-P law fits the RET data quite well for these systems, the AON works less well.
The AON is derived from a semi-classical infi- The essential step in the derivation of the AON is to assume an exponentially repulsive potential for V, As long as the kinetic energy is limited in range (thermal cell conditions for instance) this approximation is always justified.
If we denote ne range of V, by r 0 the path integral becomes x exp(-R/r 0) dR (12) in which R 0 is the distance of closest approach for a particular trajectory on the isotropic potential VO, and A is the strength of the ani sotropy.
Assuming rectilinear trajectories the integral can be analytically evaluated for head-on and large impact parameter collisions. The intermediate case can be evaluated numerically and can be shown to be within a factor of two or so of the analytic result. It is important to have the proper asymptotic behaviour.
The result is" with 3/2 a 2 (% V c ro/lu (14) where V c V(R 0) This last result is arrived at by noting that physically realistic potentials will be characterised by range parameters much smaller than R O.
where c is an integration constant given by 2 2r 0 /a. In practice the two quantities a and c are treated as free parameters in a fitting procedure. This cross section is obviously a decreasing function of n, which falls to zero for n a.
(,) The effective impact parameter is defined as the shortest distance between the line of the transferred linear momentum of the incoming particle and the centre of mass of the ellipse.
It is important to note that n > a is physically unrealistic since it would imply a negative impact parameter. Then the quantity ha Aj remax presents the maximum transferable angular momentum. Channels for which /, >a are therefore closed by setting q/O O.

Comparison of AON with power law model
Providing that the channel number is not of the same order as a, the numerical behaviour of the power law and the AON is very similar. The differences between the two are only apparent in the large Grawert channels where the angular momentum constraints become important. In that case the numerical behaviour of the AON can be reproduced by the EP law (eq. 8). Note, however that the AON does not require the setting of a th.ird independent paraneter , and that he obtains a fittin law of the form (19) o'j,.+j (2j' + 1)/(2j + 1) a(Ij' j1-1 -b) (20) for the thermally averaged cross-section. In this very simple law the parameter a is related to the zero-impact parameter distance of closest approach R 0 t h rough a (SkTITrl)l/2 ITR02 (21) Plots of statistically weighted cross-sections against IAj -I , including data for the I2(3II)-+ He system are found to be reasonably linear providin,q the initial level is small.
Deviations are attributed to the breakdown of the ES approximation, but it is interesting to speculate that better results might be obtained by taking into account the variation of a with the relative velocity.

Energy dependence
Most recent developments have concentrated on the ener,n,y dependence of the cross-sections. ; (25) The reason for this arises from the ambiguity in the ES approximation as to wether the kinetic energy in the entrance or exit channel should be used B. J. WHITAKER AND Ph. BRECHIGNAC for the calculation of a particular cross-section. The absence of an energetic threshold for the q-O crosssections make them a natural choice for the basis functions, but this choice is supported by the fact that the lowest velocity half-collision is more efficient than its high velocity counterpart for transferring angular momentum. The smaller the larger the crosssection. This reflects the increased interaction time.
The energy dependence of the fitting law may be very important, particularly if the cross-sections are measured close to threshold. As an example of this we + consider RET in the CsH(A T. + H 2 collisional system, recently measured by Ferray, Visticot, and Sayer They observed laser excited fluorescence from rotational states up to 0.75 kT away from the initially populated state. In order to deconvolute the effects of multiple collisions they used both the AON and the IOS-P to solve the coupled rate equations. They find that the IOS-P works better than the AON for 6J >7 because of the very rapid fall-off of the AON in the high j channels. The initial rate of descent is rapid so that the parameter a is small 15 increases. This leads to a larger value of a in the high 6j channels. This variation of a with the kinetic energy is further discussed below. Indeed, coming back to the hard ellipse model the maximum transferred angular momentum has been found to be AJma x 2 (AI-B1) PO (26) where A and B are the two arms of the ellipse, and PO ' -J2' pu (27) is the linear momentum of the incident particle. As long as the kinetic energy is kept constant, which was the case in the measurements by Derouard and 19 * 3 4 Sadeghi in 12 + He, He, H2, D2, AJma x scales as 1/2 (see Fig. 1). This is in agreement with the experiment and with both the IOS-EP and IOS-AON, as shown in Ref. 20. But if the energy e is changed the effective ellipse will be obtained by var.ious "cuts" of the potential "trunk" at different heights, resulting in a variation of the important quantity (A B I) as a function of e In most usual systems the isotropic potential Vo(R) is much steeper than the first anisotropic term V2(R) (as assumed in the derivation of the AON). The consequence of this situation is that (A B 1) decreases rapidly when e increases, as it is schematically s.hown in Fig. 3. In practice the exact description of the energy dependance of Aj max requires the knowledge of the potential shape. Note that, if it is known, the rather crude approximation The Na 2 + He system is an example for which this has been done. The ab initio potential surface calculated by R. Schinke et al is such that Vo(R) and V2(R) have essentially the same R dependence, which seems to be an extreme case (very "soft" molecule).
If Vo(R) and V2(R) are taken exponential with the same range r 0 the parameter (A -B I) is readily found independent of the energy (see Fig. 3 Schematic representation of a "hard" potential (left) and of a "soft" potential (right) (see text). (A1-B1) is the difference between the two arms of the effective ellipse.
Then Aj scales as At the same time the paramax meter (x, like V2(RO) is found to scale as s (see Eq. 1/2 14) so that a. (6)