Recent quantum calculations of rotationally inelastic collisions of NaK (*A*^{1}Σ^{+}) with He or Ar in a cell experiment are analyzed using semiclassical approximations valid for large quantum numbers. The results suggest a physical interpretation of *jm* → *j*′*m*′ transitions based on the vector model and lead to expressions that explicitly involve the initial and final polar angles of the angular momentum of the target molecule. The relation between the polar angle *θ* and the azimuthal quantum number *m* links the semiclassical results for the change in polar angle (*θ* → *θ*′) to quantum results for an *m* → *m*′ transition. Analytic formulas are derived that relate the location and width of peaks in the final polar angle distribution (PAD) to the *K*-dependence of the coefficients *d*_{K}(*j*, *j*′), which are proportional to tensor cross sections *σ*_{K}(*j* → *j*′). Several special cases are treated that lead to final PADs that are approximately Lorentzian or sinc functions centered at *θ*′ = *θ*. Another interesting case, “angular momentum reversal,” was observed in the calculations for He. This phenomenon, which involves a reversal of the direction of the target’s angular momentum, is shown to be associated with oscillatory behavior of the *d*_{K} for certain transitions. Finally, several strategies for obtaining the *d*_{K} coefficients from experimental data are discussed.

## I. INTRODUCTION

Collisional depolarization of rotationally excited molecules in a cell experiment is currently of great interest.^{1–27} Experiments that investigate the collisional transfer of orientation and alignment essentially explore how collisions change the direction of the molecular angular momentum vectors, and theoretical developments (reviewed in Ref. 18) have facilitated quantitative calculations. Such studies provide far more information (and a more stringent test of theory) than cross sections or rate constants for *j* → *j*′ transitions; they ultimately probe the details of state-to-state *jm* → *j*′*m*′ processes.

We recently reported quantum scattering calculations^{25,28} for He and Ar collisions with NaK at several energies (up to 0.0025 hartrees) appropriate for related experiments^{21,27} at *T* ≈ 600 K. The largest calculations included rotational levels up to *j* = 70 for Ar and 60 for He and provided the coefficients *d*_{K}(*j*, *j*′), which are proportional to tensor cross sections^{6} and are needed to determine cross sections for *jm* → *j*′*m*′ transitions. In a recent joint publication,^{27} these calculations were compared with experiments that obtained information about *m*-changing collisions between He or Ar and NaK molecules by using polarization labeling (PL) spectroscopy. The experiments^{27} utilized an optical-optical double resonance technique to create a nonzero average value of *m* for an ensemble of NaK molecules in a rovibrational level ($v$,*j*) of the *A*^{1}Σ^{+} state. By measuring the change in the average value of *m* for molecules that underwent a collisionally induced transition to the level ($v$, *j*′), Jones *et al.*^{27} determined *d*_{1}(*j*, *j*′) and *d*_{0}(*j*, *j*′). The comparison^{27} of our calculations with experiment is generally very good, but the available data do not provide coefficients *d*_{K}(*j*, *j*′) for *K* ≥ 2. Our calculations provided results for values of *K* up to more than 100, so here we investigate several features of our quantum calculations that can be compared with models appropriate in the limit of large quantum numbers. The results confirm an appealing physical interpretation of the collision process.

One model that has been invoked to characterize the distribution of final *m*′ states in *jm* → *j*′*m*′ transitions is that the polar angle $\theta =cos\u22121[m/(j+12)]$ is approximately conserved.^{6,8,22} An alternate model, based on the approximate conservation of *m* in a specific frame defined by the kinematic apse, was formulated by Khare *et al.*^{2,4} This model is exact for a classical, impulsive collision, and Brouard *et al.*^{23} found that it was very accurate in a recent analysis of crossed molecular beam experiments on NO(*X*) + Ar.

In a recent letter,^{26} we compared our quantum calculations with a semiclassical (SC) analysis based on the well-known vector model. This analysis invoked Derouard’s^{8} idea of collision-induced “tipping” or reorientation of the molecular angular momentum vector from **j** to **j**′; the primary result was a quantitative expression for the distribution of final polar angles *θ*′ in terms of the *d*_{K}(*j*, *j*′) coefficients. Integrating this polar angle distribution (PAD) leads to a formula for semiclassical cross sections *σ*_{sc}(*jm* → *j*′*m*′), in which *m* and *m*′ are treated as continuous variables. All the dynamical information about the collision, including averaging over the incoming velocity for a cell experiment, is contained in the *d*_{K}(*j*, *j*′). For the examples discussed here, we use the coefficients that we calculated quantum mechanically, but one can also determine them with classical trajectory methods.^{16}

Our preliminary results^{26} supported the idea that the polar angle *θ* is approximately conserved in certain cases. Considering our calculated quantum mechanical *d*_{K}(*j*, *j*′) as continuous functions of *K*, we noticed that the dependence was often roughly exponential, *d*_{K}(*j*, *j*′) ∝ exp(−*βK*), where *β* depended on the transition. By making several approximations, we showed that in this special case the PAD for the final *θ*′ could be expressed as a Lorentzian-like distribution, centered at *θ*′ = *θ* and with a FWHM of 2*β*.

Here we extend our original analysis. We present a rigorous derivation of the PAD for an exponential dependence of the *d*_{K}(*j*, *j*′) that clarifies the conditions necessary for the Lorentzian form. We have also derived analytic PADs for other functional forms of *d*_{K}(*j*, *j*′) that are useful for modeling the values calculated quantum mechanically. Taken together, these results enable us to correlate the location and width of peaks in the final PAD with the *K*-dependence of the *d*_{K}(*j*, *j*′) coefficients for each *j* → *j*′ transition. Because of the simple link between the semiclassical PAD and the cross sections for *jm* → *j*′*m*′ transitions, our results lead to a straightforward way to interpret experimental data and potentially to estimate the *d*_{K}.

This paper is organized as follows. Section II summarizes quantum mechanical theory and the corresponding semiclassical formulas. We also show that the semiclassical formulas that follow from the vector model are related to previous work^{3,6} by well-defined mathematical approximations. Section III presents several special cases for which the final PAD can be obtained analytically. One of these cases, a potential with a deep well, provides an explanation for recent experimental results^{27} related to K + NaK collisions. In Sec. IV, these cases are invoked to interpret selected results from our recent calculations^{25,28} of collisions of NaK (*A*^{1}Σ^{+}) with He and Ar, including an interesting class of transitions involving angular momentum reversal. Section V discusses strategies for the analysis of experimental results, and concluding remarks are given in Sec. VI.

## II. THEORY

In this section, we summarize many relevant formulas for quantum mechanical scattering and depolarization collisions and discuss their interpretation. Since the notation in the literature is not always consistent, we will explicitly relate our nomenclature to the conventions used by various other authors.

### A. Summary of quantum mechanical formulas

We start with the standard results for rotationally inelastic scattering of a structureless atom by a rigid rotator. Arthurs and Dalgarno^{29} formulated a set of coupled equations that determine the *T* matrix, which can be used to construct cross sections for *j* → *j*′ transitions,

where *k*_{j} is the channel wave number, *J* is the total angular momentum quantum number, and *l* is the orbital angular momentum quantum number of the projectile.

For the present application we consider collisions in a cell experiment in which *m*-dependent cross sections are determined. One must then average over the direction of the incident particle, which is taken to be random. Cross sections are usually written as a sum over Grawert coefficients *B*_{λ}(*j*, *j*′),^{6,30} where *λ* is related to the angular momentum transferred (see Sec. II B),

where (:::) is a 3*j* symbol, and the *B*_{λ}(*j*, *j*′) depend on the *T* matrix elements according to

where {:::} is a 6*j* symbol. The total cross sections are

where the terms (2*λ* + 1)*B*_{λ}(*j*, *j*′) correspond to the tensor opacities $Pjj\u2032\lambda $ defined by Alexander and Davis.^{6}

In many cases, it is advantageous to define cross sections for the transfer of the moments (population, orientation, alignment, etc.) of the *m* state distribution.^{5,6,8} Alexander and Davis^{6} expressed these quantities as tensor cross sections, which can be written as

where [*n*] = 2*n* + 1, and *K* = 0 denotes the isotropic cross section *σ*(*j* → *j*′). The coefficients *d*_{K}(*j*, *j*′) are given by

where {⋯} is a 6*j* symbol. In Ref. 25, we noted that Eq. (6) can be inverted using orthogonality relations for 6*j* symbols, leading to an expression for the *B*_{λ} in terms of the *d*_{K},

where *j*_{<} is the lesser of *j* and *j*′.

We have defined “symmetrized” Grawert coefficients $B\lambda (j,j\u2032)=B\lambda (j\u2032,j)$ by including the factor $\pi /kj2$ in Eq. (2) instead of in Eq. (3). The advantage of this definition is that the *B*_{λ}(*j*, *j*′) and the *d*_{K}(*j*, *j*′) then satisfy the reciprocal relationship given by Eqs. (6) and (7). These two equations clearly show that the *B*_{λ}(*j*, *j*′) and the *d*_{K}(*j*, *j*′) are alternative representations of the same information about the collision dynamics.

Several terms related to the *d*_{K} coefficients have been used in the literature. *d*_{0}(*j*, *j*′) can be defined in terms of *σ*(*j* → *j*′) by Eq. (5), and the ratios *d*_{K}(*j*, *j*′)/*d*_{0}(*j*, *j*′) were denoted multipole transfer efficiencies by Alexander and Orlikowski.^{7} We adopt the definition

of Aoiz *et al.*,^{16} who called the *a*^{(K)} polarization parameters or multipole transfer coefficients.

The discussion in this section ignores the nuclear spin **I** and possible hyperfine effects. In alkali molecules with nonzero electronic spin **S**, hyperfine effects are dominated by the **I** ⋅ **S** interaction. These effects have been considered in some studies,^{16,31} but in the present case involving a singlet electronic state of NaK the electronic spin **S** = 0. The experiments in our laboratory^{21,27} did not resolve this structure, and we do not expect such effects to play a significant role.

### B. Physical model for polar angle distribution

Figure 1 illustrates Derouard’s^{8} semiclassical interpretation of rotationally inelastic collisions, in which the initial angular momentum vector of the target **j** changes to the final vector **j**′. The magnitude of the angular momentum transferred, denoted by *λ*, can be related to the angle *α* shown in the figure, which is called the “tipping angle.” Derouard^{8} pointed out that the Grawert coefficients *B*_{λ}(*j*, *j*′) provide information about the distribution of the tipping angles between the initial and final angular momenta of the target rotator. The relation between *α* and *λ* is

By using an approximate relation between 6*j* coefficients and Legendre polynomials given by Edmonds,^{32}

and using Eq. (9) to convert the sum in Eq. (6) to an integral, one can transform the exact expressions for the *d*_{K}(*j*, *j*′) and Grawert coefficients, Eqs. (6) and (7), to their semiclassical counterparts,

where *j*_{<} = min(*j*, *j*′). Equations (11) and (12) show that the *d*_{K}(*j*, *j*′) are Legendre expansion coefficients of the continuous form of the Grawert coefficients.

By integrating both sides of Eq. (12) over sin *α dα* and invoking the relation between *d*_{0}(*j*, *j*′) and *σ*(*j* → *j*′) given in Eq. (5), one can obtain the semiclassical version of Eq. (4),

which confirms Derouard’s interpretation^{8} that the semiclassical Grawert coefficients give the distribution of tipping angles. We will denote the product *B*(*j*, *j*′; cos *α*) sin *α* as the tipping angle distribution (TAD).

What we now refer to as the TAD has also been called the “rotational tilt”^{9} or “*j*-*j*′ correlation,”^{16} and our Eq. (12) is equivalent to Eq. (2) of Aoiz *et al.*^{16} except for a constant factor. The difference arises because the integral of our TAD is related to the *σ*(*j* → *j*′) by Eq. (13); the corresponding integral of the *j*-*j*′ correlation given by Aoiz *et al.*^{16} is normalized to unity.

Figure 2 illustrates how we use the idea of the tipping angle to formulate a model for the change in polar angle as a result of a collision. The initial rotational angular momentum vector **j** precesses about the *z*-axis with a cone angle of $\theta =cos\u22121[m/(j+12)]$. (For the experiments conducted in our laboratory by Wolfe *et al.*^{21} and Jones *et al.*,^{27} the pump laser used to produce the excited states is circularly polarized, so the “physical” *z* quantization axis is the direction of propagation of the laser.) A collision tips **j** by an angle *α*, giving rise to the final rotational angular momentum **j**′, which precesses about the *z*-axis with a cone angle of *θ*′. We are interested in the probability $Pjj\u2032(\theta ,\theta \u2032)sin\theta \u2032\u2009d\theta \u2032$ that an average collision changes *θ* to a final value in the range between *θ*′ and *θ*′ + *dθ*′. The premise of our model is that $Pjj\u2032(\theta ,\theta \u2032)$ is the average value of *B*(*j*, *j*′; cos *α*) as **j** and **j**′ sweep around their circular paths in the diagram.

The details of the analysis were presented in our preliminary report;^{26} the final result is

This expression has the correct limiting behavior for the limiting case *θ* = 0. Then the tipping angle *α* = *θ*′; all the *P*_{K}(cos 0) are equal to one, and Eq. (14) reduces to Eq. (12) so that

We can also show that the total cross section for changing *j* to *j*′ is a constant times the integrated probability for changing *θ* to any *θ*′, independent of *θ*. Integrating Eq. (14) over *θ*′ and using the relationship between *σ*(*j* → *j*′) and *d*_{0}(*j*, *j*′), Eq. (5), lead to

Equation (16) is consistent with the quantum result for collisions in a cell experiment: when one averages over the direction of the incident particle, the sum over *m*′ of the cross sections for the *jm* → *j*′*m*′ transitions is independent of the initial *m*.

If one treats *m* and *m*′ as continuous variables, then one can transform the angular distribution $Pjj\u2032(\theta ,\theta \u2032)sin\theta \u2032\u2009d\theta \u2032$ into a distribution of *m*′-levels (see Ref. 26). The result is the following semiclassical expression for the change in the continuous variable *m*:

### C. Alternate derivation of $Pjj\u2032(\theta ,\theta \u2032)$

Invoking the vector model provides a clear physical interpretation of *jm* → *j*′*m*′ transitions, but the mathematical approximations involved are not obvious. This section presents an alternate and more mathematically rigorous derivation of Eqs. (14) and (17). One can start with the following expression for *jm* → *j*′*m*′ cross sections:

This formula is exact and was presented in Ref. 6; it can be derived by substituting the expression for *B*_{λ}(*j*, *j*′), Eq. (7), into the standard result, Eq. (2) and then simplifying the sums using angular momentum algebra. Following Monchick,^{3} one now invokes a relation given by Edmonds^{32} between 3*j* coefficients and Legendre polynomials, which can be written as

Using this expression one can replace the discrete quantum numbers *m* in the 3*j* coefficients by a continuous variable and thereby obtain Eqs. (14) and (17).

It is not surprising that Eq. (19) leads to the same result as the vector model. Brussaard and Tolhoek^{33} invoked the vector model for their derivation of the large-*j* behavior of 3*j* (or Clebsch-Gordan) coefficients. In fact, Eq. (19) is very reliable. Malenda *et al.*^{25} noted that it is most accurate for *K* ≪ *j*, is exact for *K* = 0 and 1, and works fairly well even for low *j* with *K* ≈ *j*. These considerations suggest that the intuitive vector model provides useful insight even for low-*j* systems that one expects to behave very quantum mechanically.

## III. ANALYTIC RESULTS FOR SPECIAL CASES AND COMPARISON WITH CALCULATIONS

We have identified several special cases that admit analytic approximations to the final PAD. These cases correspond to functional forms of *d*_{K}(*j*, *j*′) that can approximately represent our recent calculations. If *d*_{K}(*j*, *j*′) has an exponential dependence on *K*, then one can derive a near-Lorentzian PAD that is peaked at *θ*′ = *θ* when *θ* is not too close to 0° or 180°. We reported^{26} earlier that several transitions for He + NaK and Ar + NaK can be described by this model. Two other special cases are *d*_{K} = constant and *d*_{K} = (−1)^{K}. These cases can be analyzed in very similar ways and both lead to PADs that involve sinc functions. The final special case corresponds to a strong attractive interaction between the molecule and its perturber. This case leads to a near-random distribution of the direction of the final **j**′ or equivalently to a final PAD dominated by the geometrical factor sin *θ*′. We investigate a model system that illustrates this behavior, and the results are related to recent experiments^{27} and calculations.^{16}

For each analytic form of *d*_{K}(*j*, *j*′) considered, we also derive the TAD. The discrete *d*_{K}(*j*, *j*′) and the continuous TAD provide complementary ways of interpreting the underlying collision dynamics.

### A. Exponential *d*_{K}: $Pjj\u2032(\theta ,\theta \u2032)\u2009\u2009sin\theta \u2032\u2248$ Lorentzian

This section considers the special case that the *d*_{K}(*j*, *j*′) follow an exponential form:

If we substitute Eq. (20) into Eq. (14) and extend the upper limit of the sum over *K* to infinity, then we obtain a closed form expression. Changing the upper limit should be justified as long as the exponential decay constant *β* is large enough to make exp(−*βK*) very small for *K* > 2*j*_{<}. Using a result proved in Appendix A of the supplementary material [Eqs. (A1) and (A13)], one finds

where $F12\cdots \u2009$ is a Gauss hypergeometric function. We will show below that for typical values of *β*, the shape of this function depends primarily on *θ*/*β* and *θ*′/*β*. This fact enables us to show in Fig. 3, in a way nearly independent of *β*, the distribution $Pjj\u2032(\theta ,\theta \u2032)sin\theta \u2032$ of final polar angles *θ*′ for several initial values of *θ*. Except for *θ*/*β* less than about one, the distribution is peaked near *θ*′ = *θ*. (The hypergeometric function was evaluated using Forrey’s code.^{34})

Now we consider the limiting behavior of Eq. (21) for angles *θ* and *θ*′ not too close to 0° or 180°. For our systems the argument of the hypergeometric function in Eq. (21) is then typically a real, negative number with a large magnitude. In that case the asymptotic behavior is

where the first form of the asymptotic limit (the middle expression) follows from the linear transformation formulas given by Abramowitz and Stegun^{35} (see Sec. 15.3.8). We confirmed the accuracy of Eq. (22) by numerical calculation. By using Eq. (22) and assuming *β* and *θ*′ − *θ* are small, we reduced Eq. (21) to the following near-Lorentzian form:

The first factor in curly brackets in Eq. (23) is a normalized Lorentzian centered at *θ*′ = *θ* with a full width half maximum of 2*β*. This result provides justification *a posteriori* for assuming $\theta \u2032\u2212\theta \u226a1$. In addition, *β* ≪ 1 is well satisfied for the systems we have investigated.

The analytic approximation derived here may provide valuable qualitative understanding of the shape of the PAD in certain cases. In our earlier report^{26} we discussed several transitions of NaK induced by He or Ar for which *d*_{K}(*j*, *j*′) ≈ *A* exp(−*βK*). There were also cases where the *d*_{K}(*j*, *j*′) were well fit by a sum of two exponentials, and the final PAD was the sum of the corresponding two Lorentzians. However, a one- or two-exponential fit will not always be adequate. For quantitative results, the exact semiclassical formula Eq. (14) is more accurate.

### B. Exponential *d*_{K}: Tipping angle distribution

A direct evaluation of the infinite sum approximation to the TAD is given in Appendix A of the supplementary material [Eqs. (A14) and (A15)]. For small *θ*′ and *β*, the result is

We have added the superscript “is” (for infinite sum) as a reminder that the sum in Eq. (14) was extended to infinity. The expression in curly brackets, which depends only on *θ*′/*β*, determines the shape of the distribution. The normalization follows by extending the integral over *θ*′/*β* to infinity and using $\u222b0\u221ex(1\u2009+\u2009x2)\u22123/2\u2009dx=1.$ The other factors determine the height. The maximum of Eq. (24) occurs at $\theta \u2032/\beta =2/2\u22480.707$; thus *β* is a characteristic tipping angle.

The dual role of the parameter *β* in the TAD and in the exponential decay of the *d*_{K} coefficients leads us to consider the connection between these two functions. Larger values of *β* correspond to larger average tipping angles and to more rapid decay of the *d*_{K} coefficients. We can relate the relative size of *d*_{K} for small and large values of *K* to the relative effectiveness of collisions at randomizing the coarse and fine-grained structure of the distribution of *m* levels. Large changes in *m* are necessary to affect significantly the lower-*K* Legendre moments of the *m* distribution (e.g., orientation or alignment). Such changes require a large tipping angle. Conversely, the finer details of the *m*-distribution determine the higher-*K* moments; these moments are less likely to survive collisions even if the range of the TAD is short. Thus the TAD provides information about which moments of the initial *m* distribution are likely to be preserved or destroyed in typical collisions.

We now extend our analysis of the TAD to the case where the finite upper limit 2*j*_{<} of the sum in Eq. (12) is retained. The details are presented in Appendix B of the supplementary material [Eqs. (B19)–(B22)]. The result, labeled “fs” for finite sum, is

where $K0=2j<+1$. The correction term due to the finite sum is

where $\Gamma (a,z)$ denotes the incomplete gamma function. The expression in Eq. (27) is based on an asymptotic formula and is not expected to be quantitatively accurate in all cases, but it clearly suggests that oscillations that arise in the finite sum for the TAD will be damped out exponentially when the sum is extended to infinity.

Figure 4 compares several different calculations of the TADs for *j* = 29 to *j*′ = 33 transitions in He–NaK collisions. The solid circles labeled “QM” are fully quantum calculations of the discrete Grawert coefficients *B*_{λ}(*j*, *j*′) multiplied by sin *θ*′; the value of the angle *θ*′ related to each *λ* is given by Eq. (12). The solid line labeled “Exact SC” was determined using the exact semiclassical formula given by Eq. (14), with the calculated coefficients *d*_{K}(*j*, *j*′). This calculation roughly follows the quantum points, but the oscillations are not as pronounced. The dotted line labeled “SC-is” is the analytic approximation based on fitting the calculated *d*_{K}(*j*, *j*′) to the form *A* exp(−*βK*) and using Eq. (24). This approximation gives a smooth curve that passes between the quantum oscillations. Finally, the dashed curve labeled “SC-fs” was determined using Eqs. (25) and (26). The oscillations of this curve are centered on the previous curve, “SC-is.” It is worth noting that this analytic approximation to the finite sum should be quite accurate for the curve shown (*K*_{0} = 59 and *β* = 0.057), as demonstrated by Fig. 1 in Appendix B of the supplementary material. We therefore attribute the slight deviation of the “SC-fs” from the “Exact SC” curves to the difference between the exponential fit to the *d*_{K}(*j*, *j*′) and the actual calculated points. The oscillations in these two curves occur at very similar values of *θ*′/*β*.

The curves denoted SC-is and SC-fs in Fig. 4 provide the clearest illustration of the effect of extending the finite sum in Eq. (14) to infinity. [Eq. (20) is assumed in both cases.] The finite sum in the semiclassical model leads to undulations in the distribution, which vanish for the infinite sum. Such behavior can be observed in many systems when classical distributions are very close to the average of quantum oscillations.

### C. *d*_{K} = constant or (−1)^{K}: $Pjj\u2032(\theta ,\theta \u2032)\u2009\u2009sin\theta \u2032\u2248$ sinc

Let us first consider the case that *d*_{K} is a constant, *A*, for every *K*. The analysis leading to Eq. (C4) in Appendix C of the supplementary material shows that in this case

As the value of *j*_{<} increases, the distribution given by Eq. (28) becomes more and more sharply peaked. Using the integral (from Ref. 36),

one can show that

where *δ* is the Dirac *δ* function. This result explicitly confirms that the initial polar angle is conserved when *j* and *j*′ are large and the *d*_{K} are constant.

A related special case is the following:

(These forms are equivalent for integer values of *K*, but we will occasionally use the cosine form to provide a continuous interpolation.) For this case we must evaluate

If we denote the supplement of the angle *θ* by *θ*_{s}, then we can express the symmetry of the Legendre polynomials as

Using this result we can replace the sum in Eq. (32) by

which is just Eq. (C1) of the supplementary material with *θ*_{s} instead of *θ* on the right-hand side (RHS). Hence the sum in Eq. (32) can be evaluated by replacing *θ* by *θ*_{s} in Eq. (28) on the RHS, leading to

This result is rather interesting because it predicts that an oscillating *d*_{K} can cause a large change in the polar angle, from an initial value near zero to a final value near *π*. We will discuss examples of this effect in Sec. IV.

### D. *d*_{K} = constant or (−1)^{K}: Tipping angle distribution

We worked out the TAD for *d*_{K} = constant in Appendix C of the supplementary material [see Eq. (C14)]. Using the same arguments as in Sec. III C, one can show that the TAD for *d*_{K} = (−1)^{K} is given by replacing *θ*′ by its supplement $\theta s\u2032$ on the right-hand side of Eq. (C14). (One also needs to use $sin\theta \u2032=sin\theta s\u2032$.)

### E. *d*_{K} = *d*_{0}*δ*_{K0}: Random orientation of final *j*′

If *d*_{K} = *d*_{0}*δ*_{K0}, another special case arises, which can be regarded as the *β* → ∞ limit of *d*_{K}(*j*, *j*′) = *Ae*^{−βK}. In this special case, the only moment of the *m*′-distribution retained after many collisions is the population, so all *m*′ levels are equally populated, and

as well as the relationship between *σ*(*j* → *j*′) and *d*_{0}(*j*, *j*′) given by Eq. (5). The semiclassical and quantum mechanical *m*′-distributions are identical in this case, since Eq. (19) is exact for *K* = 0,

Therefore the angular distribution is

which one can confirm from Eq. (14) by noting that *P*_{0}(cos *θ*) = 1. In this case, the final **j**′ is equally likely to be oriented in any direction, and the angular distribution is dominated by the geometrical factor sin *θ*′.

We were able to approach the special case *d*_{K} = *d*_{0}*δ*_{K0} by choosing a model potential with a very deep well (on the order of 0.16 *E*_{h} or about 4.4 eV). To determine the model potential, we started with an analytic fit to a PES for LiCN provided by Essers *et al.*^{38} We started with this fit to ensure that the angular dependence and curvature of the wells in our model potential would be reasonable. We emphasize, however, that our model PES is not intended to be an accurate representation of LiCN. The analytic fit of Essers *et al.* approaches its asymptotic (large *R*) value very slowly and would lead to a lengthy scattering calculation. To reduce the amount of computer time involved, we replaced the fit provided by Essers *et al.* with a Morse potential for each angle, with the constraint that each Morse potential have the same asymptotic (large *R*) value. The details of this model potential are given in Ref. 28.

We performed scattering calculations using our deep well model PES for a total energy of *E* = 0.002 *E*_{h}. The rotational energy levels were determined from *E*_{j} = *Bj*(*j* + 1) with the rotational constant *B* = 8.655 × 10^{−6} *E*_{h} appropriate for CN. Our scattering calculations were performed such that the *d*_{K}(*j*, *j*′) were converged with respect to the different parameters involved in the calculation.

Panel (a) of Fig. 5 shows *d*_{K}(*j*, *j*′)/*d*_{0}(*j*, *j*′) for *j* = 10 and *j*′ = 12. The value for *K* = 1 is about an order of magnitude smaller than for *K* = 0 and the *d*_{K}(*j*, *j*′) for *K* ≥ 5 are negligible. These values approach the special case where *d*_{K}(*j*, *j*′) = *d*_{0}(*j*, *j*′)*δ*_{K0}, as do the *d*_{K}(*j*, *j*′) for other transitions as well. The quantum mechanical and semiclassical *m*′-distributions are very broad and nearly constant; examples are shown in panel (b) of Fig. 5. Figure 6 shows that the corresponding angular distributions for different initial *θ* are dominated by the factor sin *θ*′.

The results are just what one might expect. The attractive well is very deep, so in any collision the atom and the molecule get very close together and interact strongly, randomizing the final *m*′-levels of the molecule.

This picture can be related to other recent experiments^{27} and calculations.^{16} The data of Jones *et al.*^{27} indicate that collisions of NaK (*A*^{1}Σ^{+}) with potassium (K) typically destroyed over 90% of the initial orientation of the NaK rotational levels, corresponding to *d*_{1}/*d*_{0} ≲ 0.1 for most transitions. This result is consistent with the special case considered here of a deep attractive well. Such a potential would be plausible for the interaction of electronically excited NaK with the open shell K atom. Furthermore, Aoiz *et al.*^{16} found in their calculations for OH (*A*^{2}Σ^{+}) that the polarization parameters *a*^{(K)} = *d*_{K}/*d*_{0} were “close to zero” for all *K* > 0 and that “the OH(A) angular momentum is almost completely depolarized in a single collision.” They also attributed this behavior to the “strongly attractive” potential surface.

## IV. SELECTED RESULTS FOR He AND Ar COLLISIONS WITH NaK

The special cases considered in Sec. III provide a way of analyzing many of the features we observed in our calculations of *jm* → *j*′*m*′ transitions in collisions of He and Ar with NaK (*A*^{1}Σ^{+}). We noted in Ref. 26 that in many cases $Pjj\u2032(\theta ,\theta \u2032)sin\theta \u2032$ was approximately either a Lorentzian or a sum of two Lorentzians. We now consider additional examples where the essential features of the final PAD or TAD can be described as a composite of several special cases.

### A. *j* = 55, *m* $\u2192$ *j*′ = 57, *m*′ transitions in Ar + NaK

Many PADs for transitions involving Ar could be approximately described as the sum of a broad and a narrow Lorentzian. Here we describe one example, the 55 → 57 transition. We will discuss the behavior of the *d*_{K}(*j*, *j*′) coefficients, the PAD, and the TAD.

Figure 7 shows quantum mechanical cross sections, semiclassical PADs, and the calculated *d*_{K}(*j*, *j*′) that determine them. In order to compare discrete cross sections *σ*(*jm* → *j*′*m*′) with continuous functions $Pjj\u2032(\theta ,\theta \u2032)sin\theta \u2032$, one must scale the cross sections in some way. Because of the many common terms in Eqs. (14) and (17), we can equate

if we set $\theta =cos\u22121[m/(j+12)]$ (and do the same for *θ*′). This procedure leads to a meaningful comparison between continuous *θ*′ distributions and the quantum points [scaled according to Eq. (40) and plotted at the value of *θ*′ corresponding to each *m*′].

We chose an initial value of *m* = 39, corresponding to *θ* = 45.36°. Figure 7(b) clearly shows that the most likely final states *j*′*m*′ satisfy *θ*′ ≈ *θ*. The central peak of the final PAD is, in fact, well described by Eq. (23), a single function close to a Lorentzian. However, the enlargement of the large-angle tail of the distribution shows the importance of the second Lorentzian. As shown in Fig. 7(b), *d*_{K}/*d*_{0} drops sharply for small *K*, then changes form, and decreases more slowly. This behavior was fit by two exponentials. The final, sharper drop for the largest *K* was included in the fit but had only a small effect. Each of the exponentials in the fit shown in panel (a) is associated with a Lorentzian in panel (b). The slow exponential decay of the polarization parameters *a*^{(K)} = *d*_{K}/*d*_{0} gives the narrow peak that characterizes the PAD near the center, and the fast exponential gives the very broad component that accounts for the slower decay of the distribution at large angles, in better agreement with the quantum values. For the large quantum numbers of this example, the final *m* levels are fairly closely spaced. The discrete quantum points delineate a peak in good agreement with the continuous semiclassical approximation.

The fast and slow exponential components of the *d*_{K}(*j*, *j*′) coefficients lead to a TAD with two components as well. Figure 8 presents these two components: the fast exponential decay of the *d*_{K}(*j*, *j*′) coefficients corresponds to a long range TAD and the slow exponential corresponds to a short range TAD. This association follows from the connection between the range of the TAD and the exponential decay constant *β*; the discussion following Eq. (24) showed that the peak of the TAD occurs when $\theta \u2032=\beta /2$.

An interesting feature in Fig. 8 is that oscillations appear in the short range component of the TAD but not in the long range part. This behavior is consistent with the analysis leading to Eq. (27), which shows that the amplitude of the oscillations should be proportional to $(\beta K0)1/2exp(\u2212\beta K0)$. In this case *K*_{0} = 2*j*_{<} + 1 = 111, the short and long range *β* are 0.0274 and 0.2764, respectively, and one can estimate that the oscillations of the long range component of the TAD will be suppressed by a factor of about 10^{11}.

The two components of the TAD shown in Fig. 8 suggest that two distinct collision mechanisms may play a role in the collision dynamics. The first can only lead to a small tilt of the initial **j**, but that is sufficient to cause significant loss of high-*K* moments of the initial *m*-distribution because these moments are very sensitive to the fine details of the distribution of *m* levels. The second mechanism leads to large tipping angles and can significantly reorient the molecule. Such collisions lead to large changes in the coarse structure of the *m* distribution and therefore have a greater effect on the small-*K* moments. It is plausible to assign these mechanisms to glancing collisions and hard-core encounters, but more investigation is needed for a quantitative analysis.

### B. Angular momentum reversal

We have identified an interesting class of collision-induced transitions that involve large collision-induced changes in the direction of the angular momentum vector of the target diatom. This phenomenon was prominent in collisions of He with NaK for transitions with |Δ*j*| = 1 and small values of the initial *j*.

Figure 9 illustrates the effect for a series of transitions with *j* = 5, 11, and 22 and Δ*j* = + 1. The initial values *m* or *θ* were chosen so that *m* was an integer and *θ* was as close to 45° as possible. Panel (b1), for *j* = 5 and *j*′ = 6, presents the most dramatic illustration of angular momentum reversal. The initial *m* is 4, corresponding to *θ* = 43.34°. The largest peak of the final PAD is clearly centered near *θ*′ = *θ*, but there is a prominent peak near *θ*_{s} = *π* − *θ*. The largest quantum cross sections in this secondary peak occur at *m*′ = −4 and −5, and the continuous variable *m* corresponding to *θ*_{s} is −4.7. Visualized using the vector model, the transition from *θ* to *θ*′ = *θ*_{s} corresponds to “inverting” the initial precession of the angular momentum, by changing the sign of *m*. Alternatively, the direction of the final angular momentum vector is the reverse (within an azimuthal angle) of the initial value. As the initial value of *j* increases, the secondary peaks in panels (b2) and (b3) diminish in amplitude, suggesting that reversal becomes less likely.

We can correlate the features in the right-hand panels of Fig. 9 with the behavior of the polarization parameters *a*^{(K)}(*j*, *j*′) = *d*_{K}(*j*, *j*′)/*d*_{0}(*j*, *j*′) in the left-hand panels, which show the calculated coefficients and a fit of the form

The contributions to the final PAD from each term in this fit can be determined separately using the analysis in Sec. III. The constant term gives a sinc function centered at *θ*′ = *θ*, which accounts for the primary peak. Similarly, the oscillating component of the *d*_{K} leads to a sinc function centered at *θ*′ = *θ*_{s}, giving the secondary peak. Finally, the exponential term gives a very broad contribution to the final distribution. Although we have referred to this distribution as “near-Lorentzian,” in this case *β* is large; the distribution is broad; the factor $sin\theta \u2032/sin\theta $ in Eq. (23) introduces a significant asymmetry; and so the distribution does not appear very Lorentzian.

The progressive decrease in the secondary peaks in the right-hand panels (b1)–(b3) follows from the behavior of the oscillations in *d*_{K}/*d*_{0} shown in the corresponding panels on the left. One can clearly see that the oscillations become smaller as *j* increases. Also, as predicted by the analysis in Sec. III, the primary peaks in the right-hand panels become narrower as *j* (or *j*_{<}) increases [Eq. (28)]. Finally, as *j* increases, the “near-Lorentzian” becomes narrower and starts to look more like a Lorentzian.

Figure 10 illustrates how the trends shown in Fig. 9 are reflected in the Grawert coefficients and the TADs. The range of tipping angles (*α*_{min} to *α*_{max}) that can change an initial *θ* to *θ*′ is given by^{26}

In the present case, *θ* ≈ *π*/4 for all the transitions considered. Taking *θ*′ = *θ*_{s} = *π* − *π*/4, we see that a tipping angle range of *π*/2 to *π*, or 90° to 180° is necessary for angular momentum reversal. The top panel of Fig. 10 clearly confirms the occurrence of tipping angles in this range for an initial *j* = 5; the lower two panels show that tipping angles larger than about 90° decrease markedly as the initial value of *j* increases to 11 and then 21. The semiclassical TADs follow this trend in a general way, if one considers the average of the oscillations.

All three panels of Fig. 10 exhibit striking oscillations of the Grawert coefficients. Particularly for *j* = 5, the *B*_{λ}(5, 6) are large for odd *λ* and nearly zero for even *λ*. This behavior leads to *d*_{K}(5, 6) ∼ (−1)^{K} that we originally modeled. However, the striking correlation between angular momentum reversal and oscillatory polarization parameters does not provide an explanation of the underlying collision dynamics. Further investigation is needed to address that issue.

## V. ANALYSIS OF EXPERIMENTAL RESULTS

The correlation between the features of the *m* → *m*′ transitions and the dependence of the *d*_{K} coefficients on *K* suggests the possibility of determining these coefficients from experimental data. All the details for transitions between levels *j* and *j*′ are encapsulated in the 2*j*_{<} + 1 values of *d*_{K}(*j*, *j*′). Recent experiments^{21} have investigated *d*_{1}; here we consider how an analysis of measurements of many *m* → *m*′ cross sections could determine all the *d*_{K}.

When the quantum number *j*_{<} is not too large, a straightforward fit of experimental cross sections *σ*(*jm* → *j*′*m*′) should be feasible by using Eq. (17) or (18) and treating the *d*_{K} as adjustable parameters. Section V A formulates this method, which is a standard least squares approach. As *j*_{<} increases, and the number of fitting parameters become larger, it may be preferable to convert the measured cross sections to PADs and then seek to interpret the PADs in terms of the special cases presented in Sec. III. Section V B discusses this strategy.

### A. Treating the *d*_{K} as adjustable parameters

Let us assume we have a data set consisting of measured cross sections *σ*(*jm* → *j*′*m*′), all for the same *j* → *j*′ transition. For simplicity, we assume that only relative cross sections are available. We will relax this condition later, but for now we denote each data point by

where *N* is an unknown proportionality constant (the same for all *m* and *m*′ for fixed *j* and *j*′).

Using the exact quantum expression Eq. (18), we can write the value of the fit for each data point in the compact form

where *K*_{max} = 2*j*_{<}. Since we suppress unnecessary constants, we can identify

and

We find the best values (in the least squares sense) of the expansion coefficients *c*_{K} by minimizing

Setting *∂δ*/*∂c*_{K} = 0 for each value of *K*, one finds the following 2*j*_{<} + 1 linear equations for the unknown $cK\u2032$:

where

The sums in Eqs. (47), (49), and (50) are over all measured data points and need not include every *m* and *m*′.

The solution to these equations gives the *c*_{K}, and then the *d*_{K} follow from Eq. (45). Since we have so far considered relative cross sections, we have only determined the *d*_{K} to within an overall constant. That is enough to determine all the ratios *d*_{K}/*d*_{0}. If absolute cross sections are available, one can invoke Eq. (5) with *K* = 0 to relate the measured *σ*(*j* → *j*′) to *d*_{0} and thereby determine the correct scaling of the other *d*_{K}.

### B. Analyzing the PAD using the special cases

Our analysis in Sec. III focussed on special cases where the discrete *d*_{K} could be modeled by a continuous function of *K*, leading to an analytic form of the final PAD. In order to reverse this process, one must convert a data set of cross sections for *m* → *m*′ cross sections (all for the same *j* and *j*′) to PADs. If we make the usual connection between *m* and *θ*, Eqs. (14) and (17) imply that

If we assume that only relative cross sections have been measured and that *σ*_{sc} accurately represents the true cross section, then Eq. (51) implies that the measured *σ*(*jm* → *j*′*m*′) determine $Pjj\u2032(\theta ,\theta \u2032)$ within an overall constant.

For most of the special cases we considered [Eqs. (23), (28), and (35)], $Pjj\u2032(\theta ,\theta \u2032)sin\theta \u2032$ is proportional to a Lorentzian or a sinc function times $sin\theta \u2032/sin\theta $. It follows that in these cases, we can isolate the Lorentzian or sinc contributions by considering the angular dependence of $Pjj\u2032(\theta ,\theta \u2032)sin\theta sin\theta \u2032$. These considerations lead to the strategy of fitting angular distributions $\sigma (jm\u2192j\u2032m\u2032)sin\theta sin\theta \u2032$. If we assume that relative cross sections have been measured, the data will be of the form

where *N* is again an unknown proportionality constant, as in Eq. (43).

Table I summarizes the analytic forms for *d*_{K} that we considered, as well as the corresponding Lorentzian and sinc functions that can be used for the fit. One must assess which of these functions are likely to contribute to the angular distributions. For example, *F*_{3}(*θ*, *θ*′) is appropriate only if there are peaks for *θ*′ = *θ*_{s}. In certain cases, fitting a peak might require two Lorentzian functions *F*_{1}(*θ*, *θ*′) with different values of *β*. Depending on the functions selected and the parameters to be optimized, different mathematical techniques will be required.

d_{K}
. | $\sigma (jm\u2192j\u2032m\u2032)sin\theta sin\theta \u2032$ . |
---|---|

exp(−βK) | $F1(\theta ,\theta \u2032)=exp(\beta /2)\beta (\theta \u2032\u2212\theta )2+\beta 2$ |

1 | $F2(\theta ,\theta \u2032)=(2j<+1)\u2009sinc(2j<+1)(\theta \u2032\u2212\theta )$ |

(−1)^{K} | $F3(\theta ,\theta \u2032)=(2j<+1)\u2009sinc(2j<+1)(\theta \u2032\u2212\theta s)$ |

d_{K}
. | $\sigma (jm\u2192j\u2032m\u2032)sin\theta sin\theta \u2032$ . |
---|---|

exp(−βK) | $F1(\theta ,\theta \u2032)=exp(\beta /2)\beta (\theta \u2032\u2212\theta )2+\beta 2$ |

1 | $F2(\theta ,\theta \u2032)=(2j<+1)\u2009sinc(2j<+1)(\theta \u2032\u2212\theta )$ |

(−1)^{K} | $F3(\theta ,\theta \u2032)=(2j<+1)\u2009sinc(2j<+1)(\theta \u2032\u2212\theta s)$ |

As an example, we will consider the 5 → 6 transitions of NaK induced by He, which were discussed in Sec. IV B. Since *d*_{K}/*d*_{0} was fit to the form given by Eq. (41), we will fit the modified PADs using a linear combination of the *F*_{i}(*θ*, *θ*′), *i* = 1–3, listed in Table I,

This fit is nonlinear because in addition to the parameters *c*_{1}–*c*_{3}, one must optimize the width parameter *β* of *F*_{1}(*θ*, *θ*′). Many graphics programs can do such fits; we used gnuplot,^{39} but other codes are available.^{40}

Results of the fit are shown in Fig. 11. The cross sections [scaled according to Eq. (52)] were fit simultaneously for several different initial values of *m* or *θ*; panels (a)–(c) show three of these fits. The data are shown as discrete points at the appropriate *θ*′, and the fit is shown as a continuous curve constructed from the *F*_{i}(*θ*, *θ*′). Panel (b) corresponds to the data shown in Fig. 9, panel (b1), except for the extra factor $sin\theta sin\theta \u2032$. The fit is often higher or lower than the data, but the coefficients determined from the fit are remarkably good. Panel (d) shows the calculated *d*_{K}/*d*_{0} from Fig. 9(a1); the values of *d*_{K}/*d*_{0} determined by inverting the cross sections match very well the original fit to the calculated *d*_{K}/*d*_{0}. Additional fitting functions would improve the fit, but the four-parameter fit captures the general behavior of the nine ratios *d*_{K}/*d*_{0}.

## VI. CONCLUDING REMARKS

We have presented a semiclassical analysis of rotationally inelastic collisions of diatomic molecules with structureless particles in a cell. The formulas derived for the cross sections *σ*_{sc}(*jm* → *j*′*m*′) treat *m* as a continuous variable and depend on the *d*_{K}(*j*, *j*′) coefficients. We calculated these coefficients quantum mechanically, but they can also be obtained by classical trajectory methods.^{16} Comparison with rigorous coupled channel calculations shows that the expressions obtained are very accurate.

The analysis makes the usual identification of *j* and *m* with the polar angle $\theta =cos\u22121[m/(j+12)]$. The most important result is that when the behavior of the *d*_{K} can be represented by certain simple functions of *K* (often the case in our quantum calculations), the distribution of final *m*′ states, for a given initial *m*, can be related to a distribution of final polar angles that depends on *θ*′ − *θ*. We discussed several examples that illustrate the relation between the functional form of the *d*_{K} and the shape of the final polar angle distribution (PAD). The final PAD is usually peaked at *θ*′ = *θ*, but the width depends on the details of the *d*_{K}. An interesting anomaly, as yet unexplained, is the case of angular momentum reversal for small *j*, for which the final PAD exhibits a peak at *θ*′ = *π* − *θ*. This secondary peak is associated with an oscillatory *d*_{K}.

The present calculations have provided many examples of how the dependence of the *d*_{K} coefficients on *K* determines the distribution of final polar angles and hence of the final *m*′ states. The question then arises whether one can understand how the behavior of the *d*_{K} is determined by the details of the potential surface. As we discussed in Subsection III E, a deep potential well can be associated with a very broad distribution of final *m*′ states, but further investigation is needed to understand more complicated situations, such as the angular momentum reversal discussed in Sec. IV B.

## SUPPLEMENTARY MATERIAL

See supplementary material for appendixes containing the derivation of several mathematical results.

## ACKNOWLEDGMENTS

This work was supported by NSF under Grant No. PHY-1403060. Computational facilities at the Texas Advanced Computing Center (TACC) used for this research were supported by NSF through XSEDE resources provided by the XSEDE Science Gateways Program. T.J.P. acknowledges several helpful conversations with Heather Jaeger. A.P.H. and T.J.P. acknowledge many helpful discussions with John Huennekens.

## REFERENCES

_{z}-preserving propensities in molecular collisions: I. Quantal coupled states and classical impulsive approximations

_{z}-preserving propensities in molecular collisions: II. Close-coupling study of state-to-state differential cross sections

^{1}Σ electronic states

^{2}Π

_{1/2})

^{2}Π) + Ar/He collisions

^{2}Π)–Ar

^{2}Π) in collisions with helium

^{2S+1}Σ radicals by closed shell atoms: Theory and application to OH(A

^{2}Σ

^{+}) + Ar

_{2}gas in the nonadiabatic regime

^{1}Σ

^{+}): Transfer of population, orientation, and alignment

^{2}Σ

^{+}) + Ar