J. Phys. Chem. 1995, 99, 7407-7415
7407
Collisional Alignment of Molecular Rotation: Simple Models and Trajectory Analysis David Pullman**+ Department of Chemistry, Massachusetts Institute of Technology, Massachusetts Avenue, Cambridge, Massachusetts 02139
Bretislav Friedrich” and Dudley Herschbach” Department of Chemistry, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138 Received: August 17, 1994; In Final Form: December I , 1994@
We report simple model treatments and quasiclassical trajectory calculations dealing with collisional relaxation and alignment of molecular rotation in atom-diatom collisions. Since these models and trajectories involve single collisions, our results do not pertain directly to alignment due to supersonic expansions but nonetheless serve to elucidate some major aspects that need to be taken into account for mechanistic interpretations. By comparing alignments found for different potential functions (pairwise additive Lennard-Jones, purely repulsive inverse power, and hard ellipsoid) over a wide range of collision energies and impact parameters, we map out three qualitatively distinct regimes. These include “near-Goner” and “anti-Gorter” regimes, dominant for direct collisions at small and large impact parameters, respectively, and “pseudo-Goner” behavior found typically for low energy, large impact parameter collisions subject to soft, sticky potentials.
1. Introduction More than 50 years ago, Gorter suggested that collisions would foster alignment of the rotational angular momentum vector J of a linear molecule perpendicular to the direction v of the relative velocity vector.’ He argued that this occurs because molecules rotating with J parallel to v present a broadside target and thus suffer drastic randomizing impacts more often. Gorter’s alignment mechanism has been invoked in models for polarization of interstellar absorption lines,2 transport properties of gases,3 and many other phenomena. It has also long been recognized4 that collisional alignment in supersonic expansions might be exploited to enable molecular beam studies of steric effects. Until 5 years ago, however, clear evidence for substantial rotational alignment in supersonic beams had been found only for alkali the prevalent view was that for other species alignment was typically negligible. Recent work has now demonstrated pronounced alignment in seeded supersonic expansions of iodine,s-10 carbon dioxide,’ and oxygen molecules;I2 the current view is that collisional alignment in seeded expansions is typically appreciable. The recent studies have also revealed features incompatible with Gorter’s mechanism. Here we report simple model treatments and quasiclassical trajectory calculations dealing with collisional relaxation and alignment of molecular rotation in atom-diatom collisions, A BC, with the molecule treated as a rigid rotor. Since these models and trajectories involve single collisions, our results do not pertain directly to supersonic expansions but nonetheless serve to elucidate some major aspects that need to be taken into account for mechanistic interpretations. By comparing alignments found for different potential functions over a wide range of collision energies and impact parameters, we map out three qualitatively distinct regimes. These include “near-Gorter” and “anti-Gorter” regimes, dominant for direct collisions at small and large impact parameters, respectively, and “pseudo-Gorter”
+
Current address: Department of Chemistry, San Diego State University, San Diego, CA 92182-1030. @Abstractpublished in Advance ACS Abstracts, May 1, 1995. +
behavior found typically for low energy, large impact parameter collisions subject to soft, sticky potentials. Section 2 presents limiting impulsive and statistical models that provide explicit formulas for the second Legendre moment of the alignment angle; these are found to account qualitatively for some main features that are insensitive to the rotational state or intermolecular potentials. Section 3 outlines the trajectory computations and specifies the potential functions examined. Sections 4 and 5 present results and discussion of mechanistic aspects.
2. Simple Limiting Models for Rotational Alignment The basic distinction between direct and complex collisional
mechanism^'^ proves to be as relevant for collisional alignment as for inelastic and reactive collisions. Here we consider limiting models that provide useful approximate analytic formulas for the alignment parameter determined by most current experimental methods,I4-ls the second Legendre moment, a2 = ~ ( P ~ ( c o s Here is the angle between thefinal angular momentum J’ and the initial velocity v; the brackets indicate an average over the full probability distribution of Both in the impulsive and statistical limit, a2 can be easily evaluated; in those limits, it is related by angular momentum conservation to the distribution of 8,the angle between the initial and final relative velocity vectors. The initial and final relative velocities are designated as v and v’, respectively. These are associated with linear momenta p = mv and p‘ = mv‘, with m the reduced mass of the A BC system, and orbital angular momenta L = p x b and L’ = p’ x b’, with b and b‘ the initial and final impact parameters. The rotational angular momenta of the BC molecule before and after the collision are designated as J and J’ and the corresponding rotor states by the quantum numbers J and J’. Impulsive Model. Previous classical treatments by Beck et al.,I9 Eckelt and Korsch,20and Schinke and Korsch2I can readily be extended to the alignment moment. In the limit of direct, sudden collisions, the projection of J is conserved along the kinematic apse, defined as the difference of the final and initial relative momentum vectors, Ap = p’ - p. This follows from
x)).
x
0022-365419512099-7407$09.00/0 0 1995 American Chemical Society
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7408 J. Phys. Chem., Vol. 99, No. 19, 1995
Collisonal Alignment of Molecular Rotation 2 ,
a20 large 6
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Figure 1. (a) Kinematic relation between scattering angle and alignment of J for J = 0 S collisions. (b) Definitions and geometrical relations used to model alignment in J = 0 J’ collisions.
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Figure 2. Model calculations of a2 versus b for the impulsive model for J = 0: (a) S = 1, E = 0.031 eV; (b) S = 15, E = 0.031 eV; (c) S = 60, E = 0.155 eV. Calculations pertain to the Ar I2 system.
+
the scattering angle 8 by
the conservation of total angular momentum
$= J + L = J + b x p = J ’ + L ’ = J’+ b’ x p’ =$’ (1) Then, in the sudden limit where b
b’, we have
A J = J ’ - J = - A L = -(L’ - L ) % -b x Ap ( 2 ) Therefore, AJ is perpendicular to Ap. If we designate the projections of the initial J on Ap as p and of the final J’ as p’, then eq 2 can be written as p = p’. In tum, the spatial distribution of Ap with respect to p is govemed by the differential cross section, Z(8). In the sudden limit the alignment of molecular rotation is thus determined by I ( @ . Figure 1 illustrates the relationship for p = p’ = 0: at small scattering angles, 8 0, Ap tends to be perpendicular to p, and thus J’ is aligned parallel or antiparallel to p, whereas for 8 x, Ap is along p so that J’ is aligned perpendicular to p. If J is isotropic, these trends are expected to be more pronounced for J 0; then p 0 so that p’ 0 for all final J’. The basic notion is just that lower J-states can be more easily realigned in collisions than high J states, since a rapidly spinning top acquires gyroscopic stability. The dependence of a2 on the impact parameter b for collisions leading from the rotational ground state to any final state, J = 0 J’, is straightforward to evaluate in the impulsive limit. As noted in eq 2, angular momentum conservation implies that the projections of initial and final angular momenta on the kinematic apse, Ap, are equal, p = p‘. For J = 0, the projection is zero: p = p’= 0. Now the probability of J’ making a projection M’ on the initial momentum vector p and the projection 0 on Ap is given by the square of a Jacobi polynomial, Id;$, (a)I2, where a is the polar angle between Ap and p. For our purpose, it suffices to consider the semiclassical regimez2with cos x M’IJ‘ and
-
-
-
-
-
-
(3)
sin a =
b2+
p’ sin 19 p f 2- 2ppf COS
ep2
(5)
For elastic collisions, with p = p‘, this relation reduces to sin a = cos(8/2). The final ingredient of the model is the deflection function relating 8 to b. For simplicity, here we use the hard sphere deflection function, 8 = 2 arccos(b/R), where R is the sphere radius. This approximation, although limiting the quantitative accuracy of our model to nearly spherical potentials, reveals the essential features of alignment in the impulsive domain. Figure 2 shows plots of a2 versus b/R for different collision energies and final rotational states. The momenta p and p‘ are determined for given J’ and E by energy conservation, for any specified A BC reduced mass and BC moment of inertia (taken here as those for Ar 12). This simple model reveals the existence of both “Gorter” (a2 -= 0) and “anti-Gorter” (a2 > 0) regimes, for small and large b, respectively. Several other features are of interest. The dependence of a2 on impact parameter is insensitive to both J’ and E. Thus, in the impulsive limit the major property governing the magnitude and sign of the alignment moment is the range of impact parameters (or scattering angle) involved. Averaging over b for elastic hard-sphere collisions with the familiar “dartboard” distribution, proportional to d(nb2),is readily shown to yield a net a2 = - 5 / 8 . Statistical Model. In the limit of collisions proceeding via a long-lived statistical complex, the projection of J on Ap is no longer conserved. However, both the differential cross section and the rotational alignment are determined purely by the angular momentum disposal, largely independent of the collision dynamics. A wide class of directional properties, including the differential cross section and rotational alignment, can be evaluated in terms of Legendre moments that involve averages over various unobserved angles between angular momentum vectors.23 In particular,
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+
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The second Legendre alignment moment for J = 0 J’ transitions can then be obtained analytically as a function of sin a from
x
a2(sin a) = 5LTP2(cosX)l&o(a)12 sin dx =
From the geometry of Figure 1, the polar angle a is related to
where the carets designate unit vectors. These Legendre moments depend on only two scalar parameters, A = (U(L J)) for the incoming channel (reactants) and A‘ = (L’/(L’ S))for the outgoing channel (products). Depending on the partitioning of the angular momenta in the incoming and outgoing channels, there are four limiting cases, summarized in Table 1. For case I (A 1, A’ 0), the differential cross
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Figure 4. “Collision parameters” for A + BC nonreactive collisions.
0
See text. 0
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Figure 3. Contour diagram of a2 in the complex mode; and the comers correspond to the four special cases of angular momentum disposal; cf text and Table 1.
TABLE 1: Limiting Cases of Angular Momentum Partitioning for a Statistical Complex case
I
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0
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-V2
IV
I1
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0 0
section becomes isotropic while the rotational alignment becomes very sharp, approaching a delta function about x = 90”. For collisions nearing case I1 (A 1, A’ l), the differential cross section becomes strongly peaked, approaching l/sin 8, whereas the distribution of molecular rotation becomes isotropic (no alignment). Note that cases I and 111 involve rotationally inelastic collisions. By using Legendre moments derived from a phase space we can reduce eq 6 to an explicit formula,
- -
5
u2 = - - exp[ -q( 1 - A)“- qA’“] (7) 2 where q = 6.7 and r = 2.6. The angular momentum partitioning parameters, A and A‘, can be expressed in terms of the ratios z JqJLmax and z’ SmALfmax, respectively, where Jmp and J’,, are the most probable rotational angular momenta and L,, and L’,, the maximum contributing orbital angular momenta. This involves approximating the distribution of orbital angular momentum by P(L) = 2U(L,,,ax)2 for L 5 Lax = (a/&), where a is the integral cross section for forming the complex and A the de Broglie wavelength. By setting J = Jmp,we finally obtain A = 1 - 22 2z2 ln[(l z)/z], and for A’, an analogous expression in terms of z’. Figure 3 gives a contour map of the alignment moments obtained from eq 7 in this way. This displays a large valley, centered on case 1 limit, for which the alignment moment is negative. That is aptly termed a “pseudoGorter” domain; the sense of the alignment is the same as in the Gorter mechanism, but its origin is entirely different.
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3. Trajectory Computations
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Trajectory Program. The A BC collisions were simulated by solving Hamilton’s equations of motion, using a quasiclassical trajectory program. The code was based on formalism and computational strategy implemented by Dufp5 and reviewed by Truhlar and Muckermanz6 Analytic potentials
(specified below) were used to model the interaction of the three atoms. The states of the atoms and their time evolution were described in terms of intemal and relative Jacobi coordinates and their conjugate momenta. The number of initial values of these variables was reduced from 12 to 9 by confining the three atoms to the X Z plane of a space-fixed coordinate system X U , with Z the direction of the initial relative velocity vector; see Figure 4. Two additional variables were eliminated by treating the diatomic as a rigid, vibrationless rotor. The remaining seven variables were expressed in terms of the following “collision parameters”: (1) collision energy, E, determining the magnitude of the initial relative velocity v ; (2) rotational state, J , of the diatomic BC determining its rotational energy Erot= BJ(J 1) with B the rotational constant; (3) separation, e, between A and the center of mass of BC; (4) impact parameter, b; ( 5 ) tilt angle, of the BC intemuclear axis relative to the Z axis; (6) azimuthal angle, 4, of the BC intemuclear axis relative to the X axis; (7) azimuthal angle, 7,of the rotational angular momentum, J, of BC relative to a vector orthogonal to the molecular and the Z axis. The initial separation, eo, was chosen, by trial and error, to be large enough to make the interaction between A and BC insignificant. Initial values of E, J (and in some cases 4) were set to define a given batch of trajectories, and the remaining variables were chosen randomly. A sixth-order, variable stepsize integrator introduced by Gear2’ was used to solve the equations of motion. Importance Sampling of the Impact Parameter. Collisional alignment of J occurs over a wide range of impact parameters. In order to minimize the variance of the quantities characterizing the alignment, we implemented an importance sampling procedure for the impact parameter. The average value, (F(b)),of a function F(b) can be written as
+
c,
where Fo(b) = Fo(b@)) is the importance sampling function, b = b(j3)is a mapping of b on a random number 0 Ip 5 1, and C is a normalization constant. The integral of eq 8 has a Monte Carlo estimator
(9) which yields a minimum error for a given number of trajectories, N , if the ratio F(b@))/Fo(b@))varies slowly with ,d (Le., if F(b(j3))and the sampling function have similar dependence on p). By choosing Fo(b(B)) = l/b, the distribution of b is flat since Fo(b@))bdb = db; as a result, the probability of choosing a given b is independent of b, i.e., the probability of a trajectory
7410 J. Phys. Chem., Vol. 99, No. 19, 1995
Collisonal Alignment of Molecular Rotation
originating in a given impact parameter bin is the same for all bins. With this choice of Fo(b@)),eq 9 can be recast as
where N is the total number of trajectories whose impact parameters fall between bmin and &a,. For most of our computations N was 20 000-30 000. Evaluation of Rotational Alignment. Legendre moments can be obtained from a batch of trajectories by fitting the computed spatial distribution of rotational angular momentum to a polynomial expansion of the x distribution. However, this can be done more efficiently by using eq 10. For instance, a2 can be evaluated by setting F = P ~ ( C Ox), S i.e.
where the value x; is obtained from the ith trajectory. Higherorder Legendre moments a4 and a6 were calculated in the same fashion. Potential Energy Surfaces. The Ar I2 system was chosen as a prototype for study of collisional alignment. This choice was prompted in part by computational convenience and in part by availability of alignment data from seeded supersonic beam studies that could be compared with the theoretical results. However, both the interaction potential and kinematic parameters of the ArI2 system were varied over a wide range to elucidate their effect on the alignment. Spectroscopic data available for van der Waals complexes of Ar with homopolar halogens are generally consistent with a model where the bonding is dominated by pairwise additive forces.28 For instance, in Arc12 the distance between Ar and the center of mass of Cl2 is 3.7 A; an estimate based on van der Waals radii gives 3.8 A. Likewise, the measured well depth of A r c 1 2 is 0.023 eV, nearly twice the well depth of A r 2 , as predicted by a pairwise additive model. Therefore, the primary potential energy surface employed in our Ar I2 trajectory calculations was pairwise additive, with the van der Waals well located in the T configuration. The Ar-I bond energy was set equal to 0.0155 eV, or one-half the experimentally determined value for the ArI2 complex.28 The equilibrium distance between Ar and each I atom was approximated by the Ar-Xe equilibrium distance. The Ar-I pair was described by the LennardJones (12,6) potential (hereafter denoted L-J), and the 1-1 pair was treated as rigid, with the I-atoms at the equilibrium distance of the 12 molecule.29 Trajectory calculations also were carried out with four other potentials. One was a purely repulsive but “soft” potential, obtained by pairwise addition of L-J potentials with the attractive term removed; for this potential the r-I2 term was enhanced by an exponential factor, resulting in a dependence reproducing the shape of the repulsive branch of the L-J potential within 1%. Another potential, used for comparison, was a hard-shell ellipsoid potential. Figure 5 shows contour plots of these potential surfaces. The analytic forms of the potentials are given below. Table 2 lists values of the parameters. In addition, we used a hard-shell bispheroid potential that led to results similar to those obtained for the hard ellipsoid, and an ellipsoidal function whose location and depth of the energy well are orientation dependent. Both linear and T-shape configurations were used (corresponding to the semimajor axis along or perpendicular to the bond). This type of function, also employed by Maine and Keil,30gives rise to rotational alignment qualitatively similar to that due to the
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Figure 5. Contour plots of the potential energy surfaces. (a) pairwiseadditive Lennard-Jones (1 2,6) potential; (b) pairwise-additive purely repulsive potential; (c) hard ellipsoid potential. See text and Table 2.
pairwise-additive Lennard-Jones potential. (a) pairwise-additive Lennard-Jones (12,6) potential
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(b) pairwise-additive purely repulsive potential
(c) hard-ellipsoid potential
where e2 is the eccentricity.
(A~ B*)/A~
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Figure 6. Probability of complex formation (full line, left ordinate) and lifetime of the complex (dashed line, right ordinate) versus impact parameter b for E = 0.0031 eV and J = 15.
TABLE 2: Parameters of the Potential Enerm Surfaces type of surface parameters pairwise-additive E = 0.0155 eV Lennard-Jones u = 3.65 8, d = 2.666 8, pairwise-additive E = 0.0152 eV u = 3.65 A purely repulsive d=3.18, a = 103.44, b = 4.50752, c = 18.603,f= 3.90306 hard ellipsoid A = 5 8, (major semiaxis) B = 3.5 8, (minor semiaxis)
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b, 8, Figure 7. Dependence of a2 versus b for the pairwise-additive LennardJones (12,6) potential at E = 0.031 eV and initial J = 0, 15, and 60 (averaged over all final J' and all scattering angles).
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4. Trajectory Results and Analysis The chief aim here is comparison of the alignment moments derived from trajectories with the impulsive and statistical model results. We also briefly examine the anisotropy of rotational relaxation, an effect pertinent to recent studies of alignment in supersonic beams.8-' Alignment of Molecular Rotation. Trajectories are assigned to direct or complex collisional modes by monitoring the distance g(t) between A and the center of mass of BC as a function of time. If g(t) exhibits a single minimum (at the distance of closest approach), the collision is regarded as direct. If @(t)has multiple minima for the vibrationless trajectories, the collision is considered to involve an intermediate complex. For a L-J potential, the direct and complex modes correlate with the ratio of the initial translational energy to the potential well depth, E/€. When E/€ 2 1, which is always the case for purely repulsive potentials, the three-body Ar-I2 interaction is only appreciable on time scales smaller than the rotational period of the Ar-I2 quasimolecule, leaving little time for statistical redistribution of energy and angular momentum. When EIE -= 1, the three atoms stay together for the duration of a few rotations, long enough for some degree of redistribution. Figure 6 illustrates this situation for the L-J potential and J = 15. At E = 0.0031 eV, corresponding to E k = 0.1, the probability of forming a complex is appreciable, increasing with the impact parameter; the probability drops suddenly to zero at maximum impact parameter b = b,,,. However, as seen in Figure 6, the lifetime of the complex, tc, becomes considerable at impact parameters close to b,,,. There it approaches a value of 122,, where the rotational period zr = 7 ps has been calculated from the relation tr= 2n(Z/L). The moment of inertia Z = 7 x g cm2 was evaluated for the equilibrium configuration of the ArI, system for the rotation about the A - a ~ i s . ~For ' E k = 1, the probability of forming a complex drops to 0.1 and the
0
2
4 b,
,
4
6
8
10
A
Figure 8. Dependence of a2 versus b for the pairwise-additive LennardJones (12,6) potential at E = 0.155 eV and J = 0, 15, and 60 (averaged over all final J' and all scattering angles). lifeteime never exceeds 22,. This trend toward the sudden limit is stronger for lower J , as the centrifugal barrier becomes smaller. For a given batch of trajectories, the collision energy E and the initial J state were held constant; the initial spatial distribution of J was isotropic. The alignment is characterized by the second Legendre moment a2, although higher moments were evaluated as well. In our previous analysisg we showed that the truncated Legendre expansion, 1 azP2 (cos describes the spatial distribution of molecular rotation adequately, provided this is not sharply peaked or structured. Here we found that, for all the distributions derived from the trajectories, larger absolute values of a2 correspond to more peaked distributions. The a2 moments were computed as a function of impact parameter, b, and averaged over all final rotational states and scattering angles. The behavior of this averaged a2 moment vs b was found to match rather well the simple impulsive and statistical models over a wide domain of collision conditions and potential functions. Direct Mode. The trajectories assigned as direct mode indeed exhibit the features derived for the impulsive model. Figures 7 and 8 show the dependence of the alignment moment on the impact parameter for the L-J potential and various initial J states at collision energies of 0.031 and 0.155 eV, respectively. Typically, the alignment moment starts out negative (a2 < 0, so J l v is dominant), at low impact parameters, corresponding to large scattering angles. As the impact parameter increases,
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Figure 9. Dependence of a2 versus b for repulsive potentials: (a) pairwise-additive purely repulsive, J = 0, E = 0.031 eV; (b) pairwise-additive purely repulsive (full line) and hard ellipsoid (dashed line), J = 15, E = 0.031 eV; (c) pairwise-additive purely repulsive (full line) and hard ellipsoid (dashed line), J = 15, E = 0.155 eV; data averaged over all final J' and all scattering angles.
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Figure 10. Dependence on b of total a2, of a2 for collisions proceeding via a complex, and of initial a2 for the pairwise-additive Lennard-Jones (12,6) potential at E = 0.0031 eV and J = 15 (averaged over all final J' and all scattering angles). The distribution of initial J is isotropic if all trajectories are included.
and thus the scattering angle decreases, collisions leading to molecules with Jlv are gradually offset (a2 passes through zero) and subsequently outweighed by collisions producing JI Iv alignment (a2 > 0). As the impact parameter increases further, the alignment declines, finally reaching zero for large b. The alignment spans a larger range for lower J than for higher J states and increasingly so as the sudden limit is approached. For instance, for J = 0, -2.5 I a2 I: f l at E = 0.031 eV and -2.5 5 a2 5 4-1.5 at E = 0.155 eV whereas for J = 60, -0.25 5 a2 5 0.1 at E = 0.031 eV and -1.5 5 a2 5 +0.2 at E = 0.155 eV. Qualitatively similar features are seen in Figure 9 for the hard-shell ellipsoid and the pairwise-additive purely repulsive potential. Complex Mode. To foster the yield of trajectories exhibiting the complex mode, the collision energy was lowered to 0.003 1 eV. Figure 10 shows the dependence of a2 on b for the L-J potential at this low energy, with J = 15. Curves are shown both before (dashed) and after (solid) deleting the contribution from direct mode collisions. Again, a2 starts out negative at low b and passes through zero into positive values. However, before returning to zero at large impact parameters (b L b,,,; cc also Figure 6), the alignment moment for complex mode collisions exhibits a sharp negative dip. At low b, the complex
mode alignment is weaker than for the direct mode but is small there. The sharp negative dip resembles it since z,/z, in a2 near b,, correlates with the spike in the complex lifetime seen in Figure 6; this indicates that the corresponding complexes are statistical. The origin of the dip can be understood from simple mechanistic considerations. In order for a statistical complex to break up into products, the centrifugal barrier must be overcome. The barrier is lowest when the final orbital angular momentum L' is small. In this case L =$E J', and the final rotational angular momentum J' tends to point along the total angular momentumswhich, in tum, points preferentially perpendicular to the initial relative velocity vector v. The final J' is therefore aligned perpendicular to v, resulting in a2 < 0 for these complex mode collisions. We also examined the role of initial alignment for complex mode trajectories. Figure 10 shows the dependence of initial a2 on b (dotted line) for trajectories proceeding via a complex. Although the total distribution of initial J for all trajectories was isotropic, by virtue of the random choice of initial conditions, there is clearly appreciable anisotropy in the cross section for complex formation. This dependence exhibits a marked peak, corresponding to positive a:!,located in the range of b where the final a2 shows the dip. This indicates that in complex mode collisions J is preferentially aligned along the relative velocity v. The outcome of such collisions gives "pseudo-Gorter" alignment (a2 < 0). Anisotropic Rotational Relaxation. In our study of rotational alignment of I:! seeded in supersonic rare gas beams,s-'O we found under some conditions Jllv was the dominant alignment. On heuristic grounds, we attributed this to the anisotropy of cross section for rotational relaxation. The trajectory results offer support for this view. The heuristic argument hinges on the observation that the torque along J (necessary for the change of magnitude of J) that is exerted on the rotating molecule by its collision partner is largest for x n/2 and smallest for 0. In order to assess the anisotropy of the relaxation cross section, we ran trajectories with initial = 0" or 90". The rotational energy changes for these fixed alignments were compared for various J states and collision energies as a function of the impact parameter. At the higher collision energies, the impact parameter dependence of the rotational energy functions was quite bumpy, and it was not feasible to assess any anisotropic effect. However, a clear effect
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Jones potential, each having J = 15, and E = 0.015 eV, and b = 3 A, but differing in the initial value of Along with cos we show other quantities that help characterize the progress of the trajectories: the potential energy V, the distance e, and the current rotational state J. The trajectory in the upper panel starts with = 22”. The system point first rolls down the potential well as the atom and the diatomic approach (at an angle of 114” between e and the molecular axis), climbs up the repulsive wall, and rolls back down into the well. The interaction with the wall leads to rotational excitation so that the system point does not have enough energy to escape from the well. Rather, it travels along a nearly equipotential contour and after almost 9 ps again falls deep into the well. This is followed by a climb up the wall, rotational relaxation, another fall into the well, and finally escape toward the product asymptote. The general feature, present in all the trajectories examined (including those corresponding to a statistical complex), is that the alignment angle changes almost solely at the repulsive wall; there is only a very small contribution from the attractive well. In the case when = 90” (Le., L and J parallel or antiparallel), lower panel, the change of is very small even at the repulsive wall, in accord with Gorter’s geometric argument. However, the attractive well is not irrelevant to collisional alignment: it participates indirectly, by accelerating the system point toward the repulsive wall and thereby increasing the realignment effect. This is particularly significant at low collision energies. The trajectory analysis further indicates that changes in rotational state are also most likely induced by the repulsive wall interaction.
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Figure 11. Dependence of the net change of rotational energy, AErOt, on impact parameter b at E = 0.0031 eV and J = 15 (averaged over all final J’ and all scattering angles) for (a) the pairwise-additive Lennard-Jones (12,6) potential and (b) the pairwise-additive purely repulsive potential. 40
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5. Discussion
40
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L 4.5
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Figure 12. Time dependence of rotational J state, orientation cosine cos and potential energy V calculated for pairwise-additive LennardJones (12,6) potential at E = 0.0145 eV, J = 15, and b = 3 A. Panel a shows a trajectory with initial = 22O, panel (b) with = 90”.
x,
x
x
was seen in runs made at E = 0.0031 eV. Figure 11 shows this data for rotational energy transfer vs impact parameter for L-J and the pairwise-additive purely repulsive potential, and J = 40. Typically, throughout most of the impact parameter range, molecules starting with J l v indeed lose on average more rotational energy per collision than molecules starting with JI lv, as expected from the heuristic consideration of torque in the collisions. Time-Dependent Features of Trajectories. Instructive aspects of the collisional mechanism of rotational alignment are revealed by the time evolution of the trajectories. Figure 12 illustrates the time dependence of the alignment angle x for two typical trajectories calculated for the pairwise-additive Lennard-
By virtue of its clarity and geometrical character, the Gorter mechanism has long served as the reference model for collisional alignment of molecular rotation. Gorter reasoned that J’ should be perpendicular to v because molecules rotating with J parallel to v present a broadside target and are therefore subject to hard, randomizing impacts more often. The results of our trajectory calculations are consistent with Gorter’s mechanism, but only for small enough impact parameters. At higher impact parameters the preferential alignment of J’ is parallel (or antiparallel) to v for direct mode collisions. This “anti-Gorter” behavior is more pronounced for “softer” potentials but is evident regardless of whether the potential energy surface is hard shell, is purely repulsive, has both attractive and repulsive branches, or has a linear or T-shape geometry. The origin of the “anti-Gorter” behavior can be gleaned from the dependence of u2 on the initial orientation of the molecular axis. Figures 13 and 14 show the dependence on the azimuthal angle 4 (cf. Figure 4) of u2, 8,and J’ calculated with a pairwiseadditive L-J potential for J = 0, E = 0.03 1 eV, and b = 7.9 A; this impact parameter is large enough for the “anti-Gorter” mechanism to prevail (cf. Figure 7). In Figure 13 the initial tilt angle E between the molecular and the Z-axes is set to 90” (broadside approach) whereas in Figure 14 this angle is reduced to 60”. For small (or large) values of 4, the approach of the atom toward the molecular axis is nearly head-on. The force applied to the molecular axis is then along the Z-axis, inducing rotation of the molecule with J perpendicular to v (or the Z-axis), making u2 negative. For intermediate values of 4, between 30” and 150”, the atom approaches the molecular axis in a more “glancing” manner. The component of the force along the XY plane is then large, leading to rotation with J’ parallel to v. A related argument has been presented by Mayne and Kei130 for the case of a T-shaped ellipsoidal potential surface.
1414 J. Phys. Chem., Vol. 99, No. 19, I995
Collisonal Alignment of Molecular Rotation a propensity for alignment of J' perpendicular to v. Although its sense is the same as in Gorter's mechanism, the origin of this alignment is different; we refer to it as "pseudo-Gorter".
6. Conclusions
t
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P Figure 13. Dependence of scattering angle 8, a2, and final rotational state S on the azimuthal angle I$ calculated for pairwise-additive Lennard-Jones (12,6) potential at E = 0.031 eV. J = 0. and b = 7.9 A. The initial tilt angle 6 = 90" 15
12
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0 -3 0'
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4 Figure 14. Dependence of scattering angle 8, a2, and final rotational state S on the azimuthal angle I$ calculated for pairwise-additive Lennard-Jones (12,6)potential at E = 0.031 eV, J = 0, and b = 7.9 A. The initial tilt angle 6 = 60'. Our trajectory results suggest that this argument is valid regardless of the position, and even presence, of the potential energy well. The range of rotational excitation (of final rotational states) increases for the approach with a sideways component, Le., as 5 decreases. The opposite is true for the alignment of the induced J for the more broadside approach, the alignment reaches over the full range between -2.5 and +5; this range is reduced when 5 decreases. For smaller impact parameters, these trends are enhanced. Gorter's mechanism deals implicitly with direct collisions and is not applicable to complex-mode collisions. The alignment found by the trajectory simulations in the statistical complex limit is govemed by angular momentum disposal and can be evaluated by phase space theory. The a2 is negative, reflecting
A few chief conclusions of this study deserve emphasis. (1) The simple impulsive and statistical models prove quite useful for semiquantitative estimates and for interpretation of rotational alignment. (2) Near both limiting regimes, direct or complex, the alignment is closely related to the scattering angular distribution, although the reasons differ. For direct mode collisions: (3) preferential alignment of J perpendicular to the relative velocity is typical when the dominant scattering comes from small impact parameters, but (4)alignment of J parallel to v becomes preferred for large impact parameters. For complex mode collisions: (5) alignment of J l v is again prevalent for small impact parameters although less marked than with direct collisions, but (6) at large impact parameters the J l v propensity can become quite strong at a narrow range of impact parameters, reflecting an unusually long lifetime near the threshold for complex formation. (7) At least for low collision energies and small impact parameters, collisions with J l v produce much more rotational energy transfer than those with Jllv. (8) The direct mode trajectories, both rotationally elastic and inelastic, show a strong propensity for preserving the component of the rotational angular momentum along the kinematic apse; this is in accord with experimentI5 and with theoretical studies of variant quantization axes.32 Recent studies of alignment in supersonic expansions of molecular beams have raised new mechanistic questions, amply reviewed e l s e ~ h e r e . ~Since . ' ~ these involve multicollisional processes, suggestions derived from single-collision models are necessarily tentative. In particular, in view of (8), a succession of collisions might either weaken or enhance rotational alignment, if the apse develops a systematic trend. We note that pertinent evidence might be the recently observed striking variation with velocity of alignment for oxygen molecules seeded in supersonic rare gas beams.12 This is consistent with the notion that a2 becomes substantially more negative for the seeded molecules that have suffered the most collisions with the diluent gas and hence have reached higher velocities.
Acknowledgment. With pleasure we dedicate this paper to Mostafa El-Sayed, in admiration of his seminal contributions to physical chemistry and his uncanny ability to align molecules, bacteria, and joumal editors, authors, and referees. We thank the National Science Foundation for support. References and Notes (1) Gorter, C. J. Naturwissenschafren 1938, 26, 140. (2) Gold, T. Nature 1952, 169, 322. Purcell, E. M. Physicu 1969, 41, 100. (3) Beenakker, J. J. M.; McCourt, F. R. Ann. Rev. Phys. Chem. 1970, 2 1 , 47. (4) Sander, W. R.; Anderson, J. B. J. Phys. Chem. 1984, 88,4479 and work cited therein. (5) Sinha, M. P.; Caldwell, C. D.; Zare, R. N. J. Chem. Phys. 1974, 61, 491. (6) Visser, A. G.; Bekooy, J. P.; van der Meij, L. K.; de Vreugd, C.; Korving, J. Chem. Phys. 1976, 20, 391. (7) Rubahn, H.-G.; Toennies, J. P. J. Chem. Phys. 1988, 89, 287. (8) Pullman, D. P.; Herschbach, D. R. J. Chem. Phys. 1989,90,3881. (9) Pullman, D. P.; Friedrich, B.; Herschbach, D:R. J. Chem. Phys. 1990, 93, 3224. (10) Friedrich, B.; Pullman, D. P.; Herschbach, D. R. J. Phys. Chem. 1991, 95, 8118. ( 1 1 ) Weida, M. J.; Nesbitt, D. J. J. Chem. Phys. 1994, 100, 6372. (12) Aquilanti, V.; Ascenzi, D.; Cappelletti] D.; Pirani, F. Nature 1994, 371, 399.
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