13346
J. Phys. Chem. 1996, 100, 13346-13347
Recovery of the Intermolecular Potential from Inelastic Transfer Cross Sections M. A. Osborne and A. J. McCaffery* School of Molecular Sciences, UniVersity of Sussex, Brighton BN19QJ, Sussex, U.K. ReceiVed: April 23, 1996; In Final Form: June 18, 1996X
We report a method through which experimental rotational transfer (RT) data may be transformed to yield contours of the intermolecular potential. This “inversion” is achieved by fitting state-to-state integral cross sections for well-defined collision energies to an angular momentum (AM) transfer function which expresses the probability of direct conversion of linear to angular momentum. The parameters of this function are directly relatable to the repulsive intermolecular potential. The method is verified by recovering contours of an ab initio potential. The contour dimensions are returned with good accuracy while the average anisotropy is within 2%.
Introduction A major, but hitherto unattained, goal in molecular collision dynamics over the past 30 years has been to determine the intermolecular potential from experiment.1,2 Much hard-won data exists, yet, for want of an inversion method, most remains unrelatable to the intermolecular forces that govern dynamical behavior. Here an alternative conceptual framework is used to formulate a rotational transfer function3,4,5 from which the intermolecular potential may be recovered using state-to-state integral rotational transfer (RT) data. The new element that, for the first time, permits a successful outcome is the proposal that RT is controlled by the conversion of linear to angular momentum (AM).3 The transfer function represents the probability density for this process and we have demonstrated that this formulation reveals insights into the fundamental physics of RT.3,4 In the angular momentum (AM) model,3 transfer of linear to AM is expressed by the relation l ) µVrbn, where l is the AM generated by a collision with impact parameter bn, by partners of reduced mass µ and relative speed Vr.1 When the probability density of each variable component of this relation is included and each integrated over (within the limits of energy and AM conservation),3 an AM transfer function is obtained4 which reproduces known RT data sets to within experimental error. A corollary to this is that distribution functions for l and for Vr, obtained e.g. from experiment, could yield P(bn), the distribution function for the effective impact parameter, which may be related to the intermolecular potential. Verification of the method is less straightforward than might first appear. There is an abundance of data but few accurate potentials exist. Only in a small number of cases are both available for the same collision system. Here we choose data that are “calibrated” to a known potential. The “results” are state-to-state integral RT cross sections calculated by Schinke et al.6 for Na2-He using atom-rigid rotor scattering theory and an ab initio potential. We show that these integral cross sections may be used to reproduce contours of the repulsive intermolecular potential using a transfer function based on the AM theory.3,4 The AM Transfer Function The formulation of an AM transfer function is described in ref 4 based on the method outlined above. In ref 4, a MaxwellBoltzmann distribution of relative velocities was used as most data analysed was obtained in thermal cells. Here, a more restricted range of collision energies is considered and with a X
Abstract published in AdVance ACS Abstracts, July 15, 1996.
S0022-3654(96)01170-7 CCC: $12.00
Figure 1. Semilog plot of RT cross sections vs ∆j for Na2-He (ji ) 0 ) at collision energies Ec ) 0.05, 0.10, and 0.20 eV. Data points as crosses, squares, or triangles (from ref 6). Full lines are the predicted values using eq 1, varying the three main parameters until best fit was obtained.
very different probability distribution. In this case, a step function was used with velocities assumed to be equiprobable up to collision velocity Vc. The modified expression for probability density in angular momentum P(l) becomes
P(l) dl ) P(l|bn) P(bn) ) n [1/(µbn) - (Vth2µbn)/l]bn-γbn dbn dl (1) C ∫bmin
bint n
where bmin is the minimum impact parameter required to n provide torque at the maximum collision velocity Vc. The upper max where integration limit is defined as the smaller of bth n and bn th bn is the value of the effective impact parameter at the threshold or channel-opening velocity (Vth ) (2∆Eij/µ)1/2). The inelastic cross sections calculated by Schinke et al. are shown in Figure 1. Three collision energies 0.05, 0.10, and 0.20 eV were used, all well above the van der Waals minimum. Each of the data sets was fitted in turn with eq 1 and results are presented as full lines in Figure 1. Excellent fits are obtained. However, it is the values of the fit parameters that are of greatest significance in the context of the intermolecular potential. © 1996 American Chemical Society
Letters
J. Phys. Chem., Vol. 100, No. 32, 1996 13347
In the AM model4 three parameters are used in the transfer is related to function, each having physical significance. bmax n the anisotropy of the intermolecular potential. Anisotropies in the ab initio potential average 0.75 Å over the energy range recovered from the fits are considered. The values of bmax n 0.74, 0.74, and 0.72 Å for collision energies 0.05, 0.10, and 0.20 eV, respectively. This represents good agreement in view of the significant changes in the shapes of the calculated crosssection plots (Figure 1) for the three energies and approximations inherent in the method used to calculate the RT cross sections.6 The range parameter (γ ) does not vary greatly with energy and is 2.20 at 0.05 eV, 2.26 at 0.10 eV, and 2.40 at 0.20 eV. It may be related to the gradient of the potential at that energy which changes in a similar manner. We have shown that the ubiquitous bn-γ function4 in RT fitting is the result of averaging radial and angular anisotropies over the energy range explored by the experiment.4 The third parameter, C, converts probability density to total inelastic cross section and the values 86.85, 72.60, and 65.03 Å2 were obtained for the three collision energies. Some decrease in magnitude of C as energy increases is expected since the effective area of the potential contour decreases with increasing interaction energy. The cross section is a measure of this area. In a theoretical calculation and under real experimental conditions, there will be two equivalent trajectories that give a particular ∆j transition. These take place on opposite sides of the two-dimensional ellipse representing the potential at that energy. A representation of the physical area of each individual ellipse will therefore be given by one-half of the calculated or measured cross section. Assuming each section to be elliptical, C/2 ) πAB, where A and B are the semimajor and semiminor ) A - B. axes of the ellipse. The anisotropy is given by bmax n Solution of these two for A and B gives 2 2 (bmax x(bmax n ) + 2C/π n ) B) + 2 2
A ) B + bmax n
(2) (3)
Results For the three collision energies of the experimental data, we obtain ellipse parameters: A ) 4.75 Å, B ) 4.01 Å at 0.05 eV; A ) 4.43 Å, B ) 3.69 Å at 0.10 eV; and A ) 4.22 Å, B ) 3.49 Å at 0.20 eV. These are shown in Figure 2 in the form of elliptical contours at the three collision energies along with the ab initio potential contour plot for Na2 + He.6 The values obtained from the AM transfer function fit are very close to those of the ab initio potential. At worst the recovered values deviate by 20% from the original potential; at best they are within 8%. Note that in this analysis, we have made use of the closeness to elliptical shape of each contour of the repulsive intermolecular potential. The illustration in Figure 2 indicates that this is a reasonable approximation in the case of (A)Na2He which is likely to apply to other molecules in their ground electronic states. Very anisotropic potentials are unlikely to be well represented in such a fashion though an accurate value of the anisotropy would be expected as well as some indication, from the relative values of C and bmax n , of the extent of deviation. To summarize, the inversion process we report, which is based on best fit of the AM transfer function to experimental data, reproduces the absolute values of the potential contours with very reasonable accuracy and the anisotropy to within, on average, 2%. It should be emphasized that the recovery of the potential was not the result of advanced computational techniques; indeed the computational resources required are modest. The key element is the recasting of the theory of collision-
Figure 2. Intermolecular potential contour plots for Na2-He. Top, ab initio CI potential taken from ref 6. The contours correspond to V(r,Θ)) 0.9 × 10i eV, where i ) 0, -1, -2, -3 (solid lines) 0.0 eV (dotted line) and -0.9 × 10-4 eV (broken line). Bottom, elliptical contours obtained as described in the text. The inner contour corresponds to the classical turning point at Ec ) 0.20 eV, the middle 0.10 eV, and the outer contour is at 0.05 eV.
induced RT using an alternative conceptual framework3 based on the proposal that the process is essentially the conversion of linear momentum to angular momentum. Input data need only be (state-to-state) integral cross sections, though at well-defined collision energies. Molecular beam or velocity selected double resonance experiments8 are best suited to providing such data. Since the latter is entirely spectroscopic, the determination of the significant elements of the potential from experimental data may become relatively routine. Acknowledgment. We thank EPSRC for a research studentship to MAO and for support of this research. References and Notes (1) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics; O. U. P. Press: New York, 1987. (2) Schiffram, A; Chandler, D. W. Int. ReV. Phys. Chem. 1995, 14, 371. (3) McCaffery, A. J., AlWahabi, Z. T., Osborne, M. A.; Williams, C. J. J. Chem. Phys. 1993, 98, 4586. (4) Osborne, M. A.; McCaffery, A. J., J. Chem. Phys. 1994, 101, 5604. (5) AlWahabi, Z. T.; Besley, N. A.; McCaffery, A. J.; Osborne, M. A.; Rawi, Z, J. Chem. Phys. 1995, 102, 7945. (6) Schinke, R.; Mu¨ller, W.; Meyer, W.; McGuire, P. J. Chem. Phys. 1981, 74, 3916. (7) Osborne, M. A.; McCaffery, A. J. J. Phys. Chem. 1996, 100, 3888. (8) Collins, T. L. D.; McCaffery, A. J.; Richardson, J. P.; Wilson, R. J.; Wynn, M. J. J. Chem. Phys. 1995, 102, 4419.
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