Effect of quasiresonant dynamics on the predissociation of van der Waals molecules

Rotational and vibrational distributions of zero-temperature collisional rate coefﬁcients for atom-diatom scattering (cid:12) are used together with effective range theory to obtain lifetimes for predissociation. High-order indirect (cid:13) potential coupling in the quantum-mechanical calculation is interpreted using a simple classical picture that (cid:14) describes the quasiresonant dynamics of atom-diatom collisions by the conservation of classical action. The importance of closed channel thresholds in determining the structure of the distributions and the balance between (cid:15) momentum gap and near-resonant effects is discussed.


INTRODUCTION
The possibility to cool and trap molecules 1-11! pro- of an atom and a diatom, the internal energy of the diatom @ is converted into translational energy of the fragments.In pure vibrational predissociation where all of the diatom' @ s vibrational energy is converted into translational ener A gy, the predissociation widths are very small due to the lar B ge number of oscillations in the final-state continuum wave C function.This so-called ''momentum gap'' effect is re- duced @ when some of the diatom's vibrational energy is con- verted # to rotational energy.A near-resonant process of this kind D generally requires a large change in the rotational quan- tum 3 number of the diatom.If the van der Waals molecule has weak C anisotropy, the direct bound-continuum potential cou- pling " is very small.In this case, the higher-order indirect potential " coupling is controlling the predissociation E 21 F HG .
However, a detailed understanding of the mechanism underlying the high-order indirect potential coupling has not been given.

I
T 7 ypically, the balance between the momentum gap and near resonant effects will produce an oscillatory rotational distribution

@
for the partial predissociation widths of van der W P aals molecules that have little or no internal angular mo- mentum.

Q
It has been suggested R 21 F TS that 3 the oscillatory rota- tional 3 distributions could be a rotational rainbow phenom- enon.
A This description requires that the rotational period of the 3 diatom be longer than the time required for the fragments to 3 separate.

In
U the present work, we consider the opposite case where the 3 rotational period is short compared to the time required for the fragments to separate.For the most weakly bound state 2 of the van der Waals complex, we find that the process of ) predissociation is controlled by the same classical dy- namical quasiresonant transfer of energy that is found in ultracold  understood by considering the simplest type of perturbative 3 scheme which is often referred to as the space-fixed distortion @ r SFD s t method.Q In the zeroth-order approximation, the 3 wave functions are computed by neglecting the nondi- agonal ¥ matrix elements V if , u and the decay process is calcu- lated B using the standard rule

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The inverse of the predissociation lifetime of the most Q weakly bound state of the van der Waals complex was found to be well approximated by ~13

A
-end angular momentum of the complex is zero and the 3 vibrational stretching quantum number is the largest pos- sible 2 integer that allows the complex to be bound.The scat- tering 3 length approximation may be obtained by setting the ef A fective range parameter r ¥Ù Typically, the real part of the scattering length is much larger than 3 the imaginary part.This allows é ê j to 3 be obtained di- rectly from the zero-energy elastic scattering cross section ë íì Figure 1 shows zero-energy elastic scattering cross sections for

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analogs that were a direct result of the correlation between q « ¬ and ¥ j Ø described @ above.A major difference, however ® , between the classical and quantum calculations arises ¥ at ultracold temperatures because it is not possible to    Bessel functions in the integrand of Eq. 9 q may be expanded in a power series.The K-matrix element then behaves as m with l i f 0 3 for ultracold collisions.The zero-temperature rate coefficients are independent of k of the cross sections may be modified for systems that possess a significant long-range potential.

"
For a van der Waals potential of the form V( u the integral m 9 q n may be performed analytically.The result p is ) When the full potential is taken into account, this unphysical Y singularity is removed and the K-matrix element is unchanged from the O ( m behavior that is due to 3 the short-range part of the potential.The right-hand side of Eq. 10 is well behaved when l i ¡ l f y h¢ 3 Å and provides a modification Equation ¸11¹ shows 2 that the K-matrix element vanishes as ¯for small k x f y »º note that the hypergeometric function 2 F 1 ( l 1,5;10;2) is pure imaginary so the K-matrix element is real¼ .The inelastic scattering cross section therefore vanishes as ¯for small k x f y , u and we see that the effect of the long- range interaction is to remove two powers of k x f y from the threshold 3 behavior produced by the short-range part of the potential. "

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Therefore the zero-temperature rate coefficient is given       that may be experimentally tested with the use of trapped 3 molecules.Since the effective range theory that was used Y to relate the collisional rate coefficients to the predisso- ciation & lifetimes applies only to weakly bound states, it will be q interesting to see whether quasiresonant behavior is also found in the more deeply bound states of the van der Waals complex.

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We leave this as a subject for future investigation.
" vides # a unique opportunity to study collisions between atoms and ¥ molecules at very low translational energies $ 12-16% and ¥ may allow experimental detection of very narrow predissociation & decay widths ' 13( .Vibrational predissociation will play " an important role in the relaxation of vibrationally ex- cited & trapped molecules when the density of surrounding at- oms ) is high and may be useful in determining Feshbach reso- nance parameters for ultracold atom-molecule collisions 0 131 .Calculations have demonstrated that very efficient and specific 2 rovibrational transitions occur in the limit of zero temperature 3 4 145 .The dynamics that produce these so-called quasir 6 esonant transitions 3 can be expected to influence the predissociation " of van der Waals molecules.The 7 importance of near-resonant effects in vibrational and rotational predissociation of van der Waals molecules has long been recognized 8 17-229 .For a van der Waals molecule consisting & molecules using perturbation theory 3 has been the subject of several detailed investigations e 20,21f .The studies concluded that perturbation theory calculations & could successfully model the qualitative features of numerically g exact close-coupling results, but could not provide # results that were quantitatively accurate h 20,21i .The reason was due to the difficulty in achieving an adequate representation p of the bound state wave function and the neglect I of potential coupling between open channels.This can be q

m
is the degree of degeneracy of the final state.If the momentum transfer is large, then the final continuum wave function n will have many oscillations in the region of overlap, and ¥ the decay width will be small.In addition, the matrix element A o 1p will C be very sensitive to small variations of the bound q state wave function.Therefore a potential source of error A in perturbation theory is an inadequate representation of ) the bound state wave function q 21 F Hr .The other major source of error in perturbation theory comes & from neglecting indirect potential coupling s 20,21t .As ¢ stated in the introduction, the largest partial widths are typically 3 found for final states for which the amount of transferred momentum is small.The direct bound-continuum potential 3 coupling elements V if that 3 allow low-momentum transfer 3 , however, arise from high-order terms in a Legendre expansion A of the anisotropy of the intermolecular potential.These 7 terms are very small for weak anisotropy van der W P aals molecules suggesting that the predissociation width will C be small unless there is a substantial contribution com- ing from the indirect intermediate potential couplings.In this case, & the perturbative expression u 1v provides " a poor approximation Q to the predissociation width due to its neglect of the indirect coupling terms.It is possible to construct more sophisticated " perturbation theories such as the secular equation perturbation " theory for the open channels w SEPT s OCx that 3 are more Q accurate than the SFD method y 20 F Tz .At this stage, how- ever A , the utility in using perturbation theory to gain a quali- tative 3 understanding of the physics of predissociation be- comes & limited.A ¢ numerically exact procedure for computing the predis- sociation 2 lifetimes is to solve a set of coupled channel equations 3 for energies below threshold.The S { matrix is then di- agonalized ¥ and the eigenphase sum is differentiated with respect p to energy to obtain the resonance widths.This approach"was taken | 13} in order to establish the validity of multichannel effective range expansions for weakly bound complexes.

¥
FIG. 1. Zero-energy elastic scattering cross sections for the condition that the internal energy of the diatom is d approximately constant, or equivalently, that the momen- tum 3 gap is as small as possible.QR VR energy transfer in atom-diatom collisions at high ener A gies has been described as a series of collisionettes 25 F ¢¡ .Each collisionette resembles a separate collision that occurs when C the rapidly rotating diatom is stretched to its outer turning 3 point and is nearly collinear with the atom.Because the 3 molecules are fully stretched, collisionettes can occur for large impact parameters and produce large cross sections £ 25¤ .In between each collisionette, however, the interaction potential " decreases by several orders of magnitude.We have shown 2 ¥ 14¦ that 3 the collisionette picture needs to be modified in the T § 0 3 limit.The distinct collisionettes are replaced by a ¥ strong modulation of the interaction potential at the char- acteristic ¥ frequency of the quasiresonant transition.Whereas the 3 high-order potential coupling description is very compli- cated & and difficult to understand, the modulation of the time- dependent @ potential provides a simplified picture of the dynamics.g The interaction potential alternates between positive and ¥ negative values and often binds the atom and diatom together 3 for several successive subcollisions.Figure 4 illustrates 3 such a long-time classical collision.The number of subcollisions 2 depends on the choice of initial conditions.Since s each subcollision follows the quasiresonant rule sepa- rately, the entire collision process does not depend on the initial d conditions and therefore obeys the quasiresonant rule.
Figures 5 and 6 show zero-temperature quenching rate coefficients & as a function of initial vibrational and rotational quantum × numbers ¨and ¥ j Ø .It was shown previously © 14ª that 3 rotational distributions like those shown in Figs. 5 and 6 had classical

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FIG.

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FIG. AE 5. Quantum-mechanical calculations of the zerotemperature rate coefficients for 4 He R TS H 2 U (V , j °) as a function of the initial vibrational and rotational quantum numbers W and j °. f the respective orbital angular momenta in the initial and ¥ final channels.For a short-range potential, the spherical

¥
the respective diatomic energies in the initial and final channels.

The7I
actual momentum dependence of the zero- temperature 3 rate coefficient near threshold will depend on the 3 anisotropy of the potential energy surface.If the anisot- ropy is weak, then the coefficient C in Eq. é 11ê will C be very small 2 or could even be zero.In order to determine the k x f y -momentum dependence of the ë j Ø ì í 4î ¤ï rate coeffi- The results are given in Figs.8 and 9.For the initial ÷ ¡ø 1, j Ø ù 7 state we find that the zero-temperature rate coefthat the long-range part of the potential has a strong 2 influence on the threshold behavior.For the initial û agreement with the exact curve.Therefore we conclude that 3 the long-range part of the potential has a weak influence on ) the threshold behavior for this case.

Figures 8 and 9
Figures8 and 9 also show that a sharp spike appears in the 3 rate coefficients when the reduced mass þ sweeps 2 below 1.2 amu.This spike occurs when the most weakly bound state 2 of the van der Waals complex approaches zero energy.These 7 so-called zero-energy resonances have a strong influ- ence A on both the elastic and inelastic scattering cross sec- tions.

3¢
Figures 8 and 9 show that when the quasiresonant transition is d on the threshold of closing, the dominant quenching rate coefficient & tends to follow an exponential dependence with the 3 momentum gap.Because the total quenching rate coeffi- cient & at zero temperature is inversely related to the predisso- ciation & lifetime of the most weakly bound state Y see 2 Eq.`2 F ba dc , u

Y
the quantum-mechanical atom-molecule scattering d is ''a coarser graining rooted in the classical mechanics of ) the collision.''The lifetimes are also strongly influenced by q the proximity of closed channel thresholds.We studied the 3 analytic structure of the threshold behavior and made FIG.