On the capture cross-section for charge neutralization, recombination, photoassociation and other barrierless reactions

Chemical Physics 270 (2001) 129±132

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On the capture cross-section for charge neutralization, recombination, photoassociation and other barrierless reactions R.D. Levine a,b,* a

The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel b Department of Chemistry and Biochemistry, The University of California Los Angeles, Los Angeles CA 90095, USA Received 21 March 2001

Abstract The collision energy dependence of cross-section for reactions where the reactants attract is discussed. Examples include charge recombination A‡ ‡ B ! products, ion±molecule reactions and other curve crossing processes. The common characteristic of such processes is that on physical grounds there is a critical distance d where capture occurs. Special attention is given to the case where the critical separation is independent of energy and/or impact parameter. A modern example where this is the case is laser-induced association of atoms. The capture cross-section has the functional form r ˆ pd 2 …1 Vg …d†=ET † where for the collisions discussed the potential is attractive, Vg …d† < 0. Such a crosssection is a decreasing function of the collision energy ET . The same functional form is also useful if the potential is repulsive, Vg …d† > 0. For this well-known case, the cross-section is an increasing function of the collision energy. Ó 2001 Elsevier Science B.V. All rights reserved.

The familiar rule of thumb is that reactions without a barrier have cross-sections that decrease with increasing collision energy. This is unlike reactions with a ®nite energy threshold. The crosssections for such reactions vanishes below the threshold energy and then rises [1±4]. Neutralization reactions, where a cation and an anion recombine, A‡ ‡ B ! products are a special case of reactions where there is a particularly strong attractive potential in the entrance valley. A simple theory for the energy dependence of the capture * Address: The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Tel.: +972-2-6585260; fax: +972-2-6513742. E-mail address: ra®@fh.huji.ac.il (R.D. Levine).

cross-section was given by Mahan and Person [5] and Mahan [6]. Mahan's work on charge recombination is cited by books on reaction theory [7,8]. In a recent paper [9] Turulski et al. develop a theory for the capture cross-section of neutralization reactions. The method used by Turulski et al. is that of Mahan. Unfortunately Bruce Mahan is no longer with us. Turulski et al. are critical of previous workers but they overlooked not only the derivation of Mahan but also important reservations that were made in Ref. [5]. These reservations were further ampli®ed in the early papers of Mahan and Person [10] and Carlton and Mahan [11]. Another approach which similarly made the point that capture does not, by itself, necessarily lead to products is the seminal early work of

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Brauman, see, e.g., Ref. [12]. What both Mahan et al. and Brauman et al. emphasized is that capture by a long range attractive potential is not necessarily synonymous with the formation of products because the capture complex can dissociate back to the reactants and it may also have other products' decay channels. Therefore, a capture cross-section is but an upper bound to a reaction cross-section. A modern version of the capture problem is laser-assisted photoassociation as discussed by Gross and Dantus [13]. Two atoms are colliding in the presence of a (fairly intense) laser ®eld at a ®xed frequency. One purpose of the experiment is to make the association occur at a (more or less) well de®ned interatomic distance d. This is achieved by choosing a laser frequency that it is somewhat to the red of the resonance absorption by an isolated atom. The point is that the long range attraction in the excited state is typically stronger than that in the ground state. Therefore, the detuning to the red makes the laser light resonant when the two atoms are approaching one another (Fig. 1). In the strict Franck±Condon approximation the absorption into a bound excited state takes place when the photon energy exactly matches the electronic energy gap. This, as shown in Fig. 1, occurs at a capture distance d. As can be seen from the potential energy curves plotted, their slopes (ˆ the forces) in the capture region are not small so that the Franck±Condon approximation is quite realistic. Note that, by detuning to the red, the capture distance is larger than the van der Waals diameter of the colliding atoms. The capture distance is independent of impact parameter because the centrifugal energy term is identical in the ground and excited electronic state and the unit angular momentum brought in by the photon is negligible compared to angular momentum of the relative motion of the heavy atoms. By using a laser at the right wavelength, one has the situation that the capture distance is well de®ned and is large. Assuming that all collisions that reach the Franck±Condon window, at the separation d, absorb light, the capture cross-section is readily computed [13]. There is only one small point where care is needed and so the argument is presented in detail. When the two atoms are at the

Fig. 1. Potential energy curves for Hg ‡ Hg and Hg ‡ Hg using parameters from Table 1 of Ref. [13]. The longer range attraction and the well (depth 370 cm 1 ) in the ground state potential, Vg …R† are hardly noticeable on the energy scale shown. The vertical asymptotic separation of the two curves is the resonance excitation energy of Hg, corresponding to the 1 S0 ! 3 P1 transition. When the two Hg atoms are closer, the electronic energy gap, 4.89 eV for an isolated atom is lowered by Ve …R† Vg …R† due to the stronger attraction in the excited state. At the relative separation d the laser frequency matches the gap. Note the strong forces, of opposite sign, in the ground and excited state, at the separation d. This helps localize the absorption of the laser light to the Franck±Condon `window' where the two atoms are at the separation d apart. A vertical transition at d prepares a bound vibrational state of the excited state. The absorption can be into an unbound state only at very low collision energies when the impact parameter is signi®cantly larger than d. The electronically excited state can then be unbound if the rotational energy is high enough.

distance d apart, the kinetic energy for the approach motion is the initial collision energy ET minus the e€ective potential energy. The latter is made up of the potential energy when the system is in the ground state, Vg …R† at R ˆ d, and the kinetic energy of the rotation of the interatomic line of centers when the impact parameter is b: ET Vg …d† ET b2 =d 2 . This is the energy available along the lines of centers and as long as it is positive the two atoms can reach the separation d. One thinks of the impact parameter b as approximately the distance of closest approach. But because the distance d is to the right of the minimum of the potential, the potential Vg …d† between the two atoms

R.D. Levine / Chemical Physics 270 (2001) 129±132

is attractive. This is the common characteristic of all the collision processes that we discuss. Therefore ET Vg …d† > ET and the distance of closest approach is closer in than b. As b increases the kinetic energy along the lines of centers (ET Vg …d† ET b2 =d 2 ) decreases in a monotonic fashion. Therefore, as b increases, one will reach a point where ET Vg …d† ET b2m =d 2 ˆ 0. This is the maximal value of the impact parameter for which the atoms can get as near as d. This value is denoted as bm . If, as assumed above, all atoms that reach the separation d absorb light, the cross-section for photoassociation is Z bm rˆ 2pb db ˆ pb2m ˆ pd 2 …1 Vg …d†=ET †: …1† 0

This cross-section is a decreasing function of the collision energy ET because the potential of the associating atoms is attractive, Vg …d† < 0. An alternative form is r ˆ pd 2 …1 ‡ jVg …d†j=ET † which explicitly emphasizes the decreasing cross-section with increasing energy. The small point where care was needed is that, at low energies, the e€ective potential energy (Vg …R† ‡ ET b2 =R2 ) has a maximum as a function of R. But for a ®xed value of R, it is a monotonic increasing function of b and so the solution for bm is unique. Eq. (1) is the result of Mahan and Person [5]. For the special case when the attractive potential is Coulombic, Vg …R† ˆ Z1 Z2 =R where Z1 and Z2 are the charges on the cation and anion respectively. Eq. (1) is however generally valid under those circumstances where it is justi®ed to center attention on an (energy independent) critical separation d. For the special case of charge neutralization, the critical distance d is determined from the point where the Coulombic potential has an (avoided) intersection with a covalent curve that leads to the products. At longer distances the covalent curve is weakly R dependent as compared to the steep fall of the ionic curve so that the intersection is at ECoulomb Z1 Z2 =d ˆ Ecovalent or [2] d ˆ Z1 Z2 =DE:

…2†

ECoulomb and Ecovalent are the asymptotic values of the ionic and covalent potentials and DE ˆ

131

 ˆ ECoulomb Ecovalent . In practical units [2], d …A† 14:35Z1 Z2 =DE (eV) where Z1 and Z2 are the number of charges on the cation and anion. Note that in this argument we have neglected the magnitude of jVe …d†j in comparison with the Coulombic potential. Here the subscript e denotes the covalent potential of the neutral reactants, that crosses the ionic curve. The strict result is that d is the solution of the implicit equation ECoulomb Z1 Z2 =d ˆ Ecovalent ‡ Ve …d†. Eq. (2) has the same dependence on Z1 and Z2 as the result of Turulski et al. [9, Eq. (15)] but otherwise has an essential di€erence. The quantity DE is not the energy di€erence between reactants and products. Rather, as introduced above, DE is de®ned in the entrance valley. It is the di€erence between the threshold energy for the formation of ions and the threshold energy for the formation of neutral reactants. The reason, as clearly discussed by Mahan et al., by others before them [14], and by many authors since, is that the critical distance is determined by a curve crossing. It is often not clear which covalent state of the reactants is important for the capture. It may be an excited electronic state and it is likely to be so whenever the electronic energy gap is large because a more nearly resonant charge transfer is more probable. A special case is curve crossing to a covalent state in which the reactants are in their ground electronic state. Then DE is threshold energy for ion pair formation starting with ground state reactants, A ‡ B ! A‡ ‡ B . In this case it follows that the magnitude of the ground state potential at the large crossing distance d is often small so that the capture cross-section is approximated by r ˆ pd 2 . Another context in which capture considerations can be useful is when the potential between the reaction products is attractive. This puts a cuto€ on the allowed range of impact parameters in the exit channel. Because this range decreases with increasing energy one can reach an energy such that the bottleneck to reaction is in the exit channel. The cross-section for reaction in the forward direction will ®rst rise, as expected for reactions with a real barrier in the entrance channel. At energies well above the barrier the other bottleneck becomes dominant and the cross-section declines

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action which is implied by the considerations of Harris and Herscbach. The origin of the energy dependence of the reaction cross-section for K ‡ CH3 I remains therefore a subject of active discussion [18]. Finally there is the opposite case of ion±molecule processes where the reaction crosssection initially decreases with increasing energy and then turns over and starts to increase.

Acknowledgements This work was supported by the Alexander von Humboldt Foundation. Fig. 2. The reaction cross-section for K ‡ CH3 I ! KI ‡ CH3 vs. the collision energy as given by the model of Harris and Herschbach [15]. The solid line is the familiar result for reaction with a barrier r ˆ pd 2 …1 E0 =ET †. This is Eq. (1) except that here the potential in the entrance channel has a barrier at the critical distance d, E0 ˆ Vg …d† > 0. The plot is for E0 ˆ  The origin of the critical distance is 1 kcal mol 1 and d ˆ 4:5 A. here too due to curve crossing [19] except that the crossing occurs when the ground state potential is already repulsive [18]. The dashed line is a capture due to the attractive potential for R < d. Harris and Herschbach used the attractive potential 3:2…d=R†2 kcal mol 1 . As is the case in general, the capture cross-section for this potential is a decreasing function of the collision energy. It is shown as a dashed line. Taking the reaction cross-section to be the lower of the two curves accounts well for the measured [17] cross-section. Harris and Herschbach obtained a more quantitative ®t to experiment by multiplying the cross-section by a steric factor of 0.8.

as shown in Fig. 2. This is a special case of a transition state switching. A simple capture model that mimics such an energy dependence is to assume that the maximal impact parameter is the smaller of the limits imposed by the entrance and exit channels. An early application of such a model was to the K ‡ CH3 I reaction [15]. Harris and Herscbach used a potentials suggested by an optical model analysis [16] and qualitatively the resulting cross-section (Fig. 2) is rather similar to the measured one [17]. So far, trajectory computations run on more realistic surfaces have failed to reproduce the transition state switching for this re-

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