Application of Heisenberg's S Matrix Program to the Angular

Article pubs.acs.org/JPCA

Application of Heisenberg’s S Matrix Program to the Angular Scattering of the H + D2(vi = 0, ji = 0) → HD(vf = 3, jf = 0) + D Reaction: Piecewise S Matrix Elements Using Linear, Quadratic, Step-Function, and Top-Hat Parametrizations Xiao Shan† and J. N. L. Connor* School of Chemistry, The University of Manchester, Manchester M13 9PL, England ABSTRACT: A previous paper by Shan and Connor (Phys. Chem. Chem. Phys. 2011, 13, 8392) reported the surprising result that four simple parametrized S matrices can reproduce the forward-angle glory scattering of the H + D2(vi=0,ji=0) → HD(vf=3,jf=0) + D reaction, whose differential cross section (DCS) had been computed in a state-of-the-art scattering calculation for a state-of-the-art potential energy surface. Here, v and j are vibrational and rotational quantum numbers, respectively, and the translational energy is 1.81 eV. This paper asks the question: Can we replace the analytic functions (of class Cω) used by Shan−Connor with simpler mathematical functions and still reproduce the forward-angle glory scattering? We first construct S matrix elements (of class C0) using a quadratic phase and a piecewise-continuous pre-exponential factor consisting of three pieces. Two of the pieces are constants, with one taking the value N (a real normalization constant) at small values of the total angular momentum number, J; the other piece has the value 0 at large J. These two pieces are joined at intermediate values of J by either a straight line, giving rise to the linear parametrization (denoted param L), or a quadratic curve, which defines the quadratic parametrization (param Q). We find that both param L and param Q can reproduce the glory scattering for center-of-mass reactive scattering angles, θR ≲ 30°. Second, we use a piecewise-discontinuous pre-exponential factor and a quadratic phase, giving rise to a step-function parametrization (param SF) and a top-hat parametrization (param TH). We find that both param SF and param TH can reproduce the forward-angle scattering, even though these class C−1 parametrizations are usually considered too simplistic to be useful for calculations of DCSs. We find that an ultrasimplistic param THz, which is param TH with a phase of zero, can also reproduce the glory scattering at forward angles. The S matrix elements for param THz are real and consist of five nonzero equal values, given by S̃J = 0.02266, for the window, J = 21(1)25. Param THz is sufficiently simple that we can derive closed forms for the partial wave scattering amplitude, f(θR), and the near-side (N) and far-side (F) subamplitudes. We show that window representations of f(θR) provide important insights into the range of J values that contribute to the reaction dynamics. Other theoretical techniques used are NF theory for the analysis of DCSs and full and NF local angular momentum theory, in both cases including up to three resummations of f(θR) before making the NF decomposition. Finally, we investigate the accuracy of various semiclassical glory theories for the DCS of param L. By varying one phase parameter for param L, we show that the uniform semiclassical approximation is accurate from θR = 0° to close to θR = 180°. Our approach is an example of a “weak” form of Heisenberg’s S matrix program, which does not use a potential energy surface(s); rather it focuses on the properties of the S matrix. Our method is easy to apply to DCSs from experimental measurements or from computer simulations.

1. INTRODUCTION Understanding the dynamics of state-to-state chemical reactions is a topic of fundamental importance in physical chemistry. The standard procedure for the first-principles calculation of collisional observables for a transition from known initial states to known final states uses the following scheme,

The Royal Road, has resulted in impressive advances in the understanding of the dynamics of chemical reactions, as reviewed recently by Hu and Schatz.2 However, the flipside to this progress has also been pointed out by Hu and Schatz: Theorists have often taken decades to understand the wealth of data generated by experiments, and new computations have frequently resulted in “confusions” in their interpretation.2 Consider, for example, the F + H2 → FH + H reaction, as measured in the famous crossed molecular-beam experiments

initial states → potential energy surface(s) → S matrix → final states

(1) Special Issue: Jörn Manz Festschrift

We have called scheme 1, The Royal Road of Reaction Dynamics in ref 1, because of its importance and ubiquity. As a result of rapid progress in computer technology and scattering algorithms during the past 30 years or so, research following © 2012 American Chemical Society

Received: June 29, 2012 Revised: August 8, 2012 Published: August 9, 2012 11414

dx.doi.org/10.1021/jp306435t | J. Phys. Chem. A 2012, 116, 11414−11426

The Journal of Physical Chemistry A

Article

of Neumark et al.3 It took 19 years before the forward peak in the small angle scattering of the FH(vf=3) vibrational state was recognized as a glory.4,5 Another example concerns two isotopologs of the fundamental H + H2 → H2 + H reaction: Mielke et al.6 report that it took 75 years to bring experimental and theoretical thermal rate coefficients into agreement. One reason for this slow progress along The Royal Road is the difficulty in calculating potential energy surfaces of sufficient accuracy for use in collisional calculations.1,2 Another difficulty concerns the numerical scattering (S) matrix elements generated in large-scale computations. Under semiclassical conditions, they often consist of a long list of complex numbers, which can be hard to understand and interpret.1 Also a new scattering computation is required for a new potential energy surface and a new reaction. In a companion paper1 to the present one (hereafter denoted SC), we have investigated a complementary, simpler, approach to understanding the angular scattering of state-to-state reactions, which is inspired by Heisenberg’s S matrix program.7−10 This alternative approach can be summarized by the scheme initial states → S matrix → final states

used. Here analyticity is with respect to J, the total angular momentum variable. The purpose of this paper is to ask the question: Using the same design principles as SC, can we replace the analytic f unctions with simpler mathematical f unctions and still reproduce the forward-angle scattering of the H + D2 reaction? To investigate this question, we first use S matrix elements constructed from a quadratic phase and a piecewise-continuous pre-exponential factor consisting of three pieces. Two of the pieces are constants, with one taking the value N (a real normalization constant) at small J; the other piece has the value 0 at large J. These two pieces are joined at intermediate values of J by either a straight line, giving rise to the linear parametrization (denoted param L or pL), or a quadratic curve, which defines the quadratic parametrization (denoted param Q or pQ). Second, we use a limiting case of pL and pQ that gives rise to a piecewise-discontinuous step-f unction parametrization (denoted param SF or pSF); it has a discontinuity (step) of value N. We also employ a piecewise-discontinuous top-hat parametrization (denoted param TH or pTH) that has two discontinuities (steps), both of value N. The simplest parametrized function we use is pTH with its phase set equal to zero (denoted param THz or pTHz). All the parameters in our parametrizations have a clear physical interpretation. We calculate DCSs for all parametrizations using the partial wave series (PWS) representation of the scattering amplitude. To understand structure in the DCSs at small, as well as at larger, angles, we decompose the PWS scattering amplitude into near-side and far-side (N and F) subamplitudes.1,4,5,18−41 This also lets us carry out full and NF local angular momentum (LAM) analyses1,19,21,23,35−41 of the scattering dynamics. In addition, we perform up to three resummations of the PWS scattering amplitude1,20,21,23,25,34−37,39−41 before making the NF decomposition, as this can provide an improved physical understanding of structure in the DCS. The THz parametrization is so simple that we can evaluate the full and N,F PWS DCSs in closed form. Finally, we investigate the accuracy of semiclassical glory theories1,4,5,18,19,23,38,41 for the DCS of param L (the results for param Q are similar). By varying one phase parameter for param L, we can test the glory theories for angles from θR = 0° to close to θR = 180°. We need to classify the continuity and differentiability properties of the parametrizations used in this paper; this is done in section 2. The design principles and construction of the parametrized S matrix elements are described in section 3, and section 4 outlines the partial wave and semiclassical scattering theories we require for the calculation of the full and N,F DCSs and the full and N,F LAMs. The values of the S matrix parameters for the H + D2 reaction are reported in section 5. Our partial wave and semiclassical results are then described and discussed in sections 6 and 7, respectively. Conclusions are in section 8.

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Scheme 2 does not use a potential energy surface(s); rather it is based on Heisenberg’s fundamental insight7−10 that the S matrix, in principle, contains all the information needed to calculate collisional observables. Heisenberg’s hope was to determine the S matrix using general physical principles, e.g., unitarity, causality, analyticity. This hope has never been realized. To avoid this difficulty, SC used a “weak” form of Heisenberg’s S matrix program, in which four general physical principles relevant to chemical reactions were employed to suggest simple parametrized forms for the S matrix.1 The unknown parameters were then found by fitting the differential cross section (DCS) calculated from the parametrized S matrix to the DCS from a computer simulation, thereby providing insight into the reaction dynamics.1 Alternatively, experimental DCSs could be used for the fitting. Further information on Heisenberg’s S matrix program, including historical remarks, is given by SC. The state-to-state DCS studied by SC was for the fundamental reaction H + D2 (vi=0,ji =0) → HD(vf =3,jf =0) + D

at a translational energy of 1.81 eV (total energy = 2.00 eV), where v and j are vibrational and rotational quantum numbers, respectively. The subscripts “i” and “f” denote the initial and final states, respectively. The DCS for this reaction exhibits rich dynamics with a surprising forward peak that arises from glory scattering,1,4,5,11−13 accompanied by oscillations whose origin is near-side−far-side interference.1,4,5,11 The H + D2 reaction has been studied in state-of-the-art experiments by the Zare group (see refs 14−17 and earlier measurements cited in these papers). There have also been many theoretical studies; recent references can be found in SC. Other research from Manchester on this reaction has examined aspects of its dynamics in both the energy domain18−23 and the time-domain.22−25 An unexpected finding by SC was that four simple S matrix parametrizations could accurately reproduce the structured DCS for θR ≲ 30°, where θR is the center-of-mass reactive scattering angle. Each parametrization consisted of a monotonically decreasing analytic function for the pre-exponential factor and a quadratic phase, although the analyticity property was not

2. NOTATION FOR THE CONTINUITY AND DIFFERENTIABILITY OF FUNCTIONS To classify our parametrizations in terms of their mathematical simplicity, we recall the standard notation42 used for the classification of the continuity and differentiability properties of functions: C0: class of functions that are continuous. C1: class of functions that are continuous and whose first derivative exists and is continuous. 11415



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⋮ Cn: class of functions that are continuous and whose first, second, ..., nth derivatives exist and are continuous. ⋮ C∞: class of functions that have continuous derivatives of all orders. Functions in this class are sometimes called smooth. Cω: class of functions that are smooth and also analytic. In addition, C0 ⊃ C1 ... ⊃ Cn ... ⊃ C∞ ⊃ Cω, where the inclusions are strict. To the above list can be added:42 C−1: class of functions with discontinuities

that the J-shifted Eckart parametrization for the S matrix is also of the form (3), with the important difference that s̃(J) is complex valued.11,18−20 3A. Parametrization of ϕ̃ (J). As in SC, ϕ̃ (J) is chosen to be a quadratic phase, namely1 ϕ(̃ J ) ≡ ϕ(̃ a ,b ,c ;J ) = aJ 2 + bJ + c

where a, b, and c are three (real) phase parameters with a < 0 (note: it is convenient to include c, even though its value does not influence the DCS). As previously, the physical significance of a and b follows from consideration of the quantum deflection f unction, Θ̃(J), defined by1,4,5 d arg S(̃ J ) dϕ(̃ J ) Θ̃(J ) ≡ = = 2aJ + b dJ dJ

3. PARAMETRIZED S MATRIX ELEMENTS In this section, we follow the design strategy of SC for the S matrix. Four simple parametrizations for the S matrix are described, which are then used in sections 6 and 7 to calculate the DCS for the H + D2 reaction. Note that no potential energy surface is used in the construction of the S matrices. The following notation is used: We consider a (modified) S matrix element for a transition i → f and write S̃f←i J , or more simply, S̃J, where J = 0, 1, 2, ..., is the total angular momentum quantum number. When S̃J is continued to real values of J, we write S̃(J). The following four general physical principles are used to construct the S matrix elements:1 1. The forces responsible for chemical reactions are short ranged, of the order of 10−10 m. This implies SJ̃ → 0 as J → ∞. In practice, there is a maximum value of J, denoted Jmax, beyond which partial waves make a negligible numerical contribution to the PWS. N.b., this principle excludes reactions that are asymptotically Coulombic, for which the PWS is divergent. 2. Conservation of probability holds. This implies the S matrix is unitary with 0 ≤ |S̃J| ≤ 1. 3. Under semiclassical conditions, namely Jmax ≫ 1, we can continue S̃J to a. a piecewise-continuous function, S̃(J) (of class C0), with simple properties for the pieces, e.g., the preexponential factor for each piece is a constant or is monotonically decreasing or b. a piecewise-discontinuous function, S̃(J) (of class C−1), with simple properties, e.g., the pieces have constant pre-exponential factors, separated by one or two jump discontinuities (steps). 4. In the classical limit, we require a head-on collision to correspond to backward (or rebound) scattering of the products. Principles 1, 2, and 4 are the same as in SC. Principle 3 is different (weaker) compared to SC. Previously it was assumed1 that S̃(J) is analytic, i.e., of class Cω, with a monotonically decreasing pre-exponential factor, whereas now we allow S̃(J) to be of class C0 or C−1. Keeping the above physical principles in mind, we parametrize S̃(J) in polar form S(̃ J ) = s (̃ J ) exp[iϕ(̃ J )]

(4)

(5)

Equation 5 shows that Θ̃(J=0) = b; in practice, we have set b = π in all except one of our parametrizations, so that in a classical picture, a head-on collision corresponds to backward or rebound scattering of the products (see principle 4 above). The exception is the THz parametrization for which ϕ̃ (J) ≡ 0. We also have from eq 5 that the stationary-phase condition, Θ̃(J) = 0, is satisfied by J = Jg = b/(−2a), where Jg is the value of the glory angular momentum variable (which is nonintegral in general); evidently a must be negative, a < 0, because Jg is positive. 3B. Piecewise-Continuous Parametrizations of s̃(J). The pre-exponential factor, s̃(J), is parametrized as a piecewisecontinuous function of J of class C0 (but not C1) with the following general properties as J increases: (1) s̃(J) is equal to a normalization constant, N, from J = 0 to J = JN, where 0 ≤ N ≤ 1. (2) s̃(J) decreases from J = JN to J = Jz where s̃(Jz) = 0. (3) s̃(J) remains at zero for all J ≥ Jz. Note that JN and Jz are nonintegral in general. Two simple parametrized forms are used for s̃(J) whereby it decreases from N to 0 either linearly38 or quadratically in J. For both parametrizations, s̃(J) depends on three independent parameters. a. Linear Parametrization (param L). This parametrization has three pieces. It is given by ⎧N for 0 ≤ J ≤ JN ⎪ ⎪ s L̃ (J ) = ⎨ d(J − Jz ) = |d|(Jz − J ) with d < 0 and JN ≤ J ≤ Jz ⎪ ⎪0 for J ≥ Jz ⎩ (pL)

where, from the properties of a straight line, JN = Jz + N /d = Jz − N /|d|

with d < 0

(6)

In practice, we specify the three parameters N, d, and Jz, and then calculate JN using eq 6. Jz acts as the “cut-off” value of J and |d| has the effect of a “diffuseness” parameter, which characterizes how quickly s̃(J) decreases from N to 0. b. Quadratic Parametrization (param Q). This parametrization also has three pieces and is defined by

(3)

⎧N for 0 ≤ J ≤ JN ⎪ ⎪ sQ̃ (J ) = ⎨ q(J − Jz )2 with q > 0 and JN ≤ J ≤ Jz ⎪ ⎪0 for J ≥ Jz ⎩

where ϕ̃ (J) is a real scattering phase and s̃(J) is positive or zero, with 0 ≤ s̃(J) ≤ 1 and s̃(J) → 0 for J → ∞. We will see in sections 3B and 3C that s̃(J) contains a real normalization constant, N; previously we were able to extract N as a multiplicative factor.1 Also we have ϕ̃ (J) = arg S̃(J). We note 11416

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indicate the reactive scattering angle, i.e., the angle between the incoming A atom and the outgoing AB molecule in the centerof-mass collision system. 4A. Partial Wave Theory. We can expand the scattering amplitude, f(θR), in a basis set of Legendre polynomials because both helicity quantum numbers are equal to zero. We write

where, from the properties of a quadratic curve,

JN = Jz −

N /q

(7)

with q acting as the “diffuseness” parameter. Similar to param L, we calculate JN from eq 7, having first specified the three parameters N, q, and Jz. 3C. Piecewise-Discontinuous Parametrizations of s̃(J). Here we define two parametrizations for s̃(J) that are of class C−1, with one or two jump discontinuities (steps). a. Step-Function Parametrization (param SF). Our stepfunction parametrization is of standard type. It can be obtained from param L or param Q in the limit Jz → JN. It consists of two pieces. With JN = Jz, we have ⎧ for 0 ≤ J ≤ JN ⎪N ⎨ sSF J = ( ) ̃ ⎪ ⎩ 0 for J > JN

f (θR ) =

It possesses two jump discontinuities (steps), both of value N, and depends on three parameters: N, JM, and JN. We also consider a special case of pTH in which the phase is set equal to zero, ϕ̃ (J) ≡ 0; we then use the notation pTHz. Param TH and param THz are both examples of window representations for the S matrix.40,43,44 Consider a PWS with nonzero partial waves in the range J = 0 to J = Jmax; then there is a “window” if only a small range of partial waves contribute significantly to the scattering amplitude. In our case, the window in J-space is JM ≤ J ≤ JN. N.b., param TH and param THz must not be confused with terahertz or the film Top Hat. Also param THz is sometimes called a rectangular f unction or a boxcar f unction. For all the parametrizations, the full S matrix element is given by eq 3. We will attach the labels, L, Q, SF, and TH to S̃(J) and ϕ̃ (J) to indicate that the corresponding parametrization for s̃(J) is being used, i.e.,

f (θR ) = f (N) (θR ) + f (F) (θR )

(12)

with f (N,F) (θR ) =

1 2ik

∞

∑ (2J + 1)SJ̃ Q J(N,F)(cos θR ) J=0

(13)

where the Q(N,F) (cos θR) are traveling Legendre f unctions J defined by (for θR ≠ 0, π)

(8)

Q J(N,F)(cos θR ) =

Also ̃ (J ) = s TH ̃ ( J )| STHz ̃ (J ) = |STH

(11)

Under semiclassical conditions, the PWS, defined by eq 10, will contain many numerically significant terms, typically of order Jmax ≈ kR, where R is the reaction radius. As well as PWS calculations of DCSs, we have also used the following PWS techniques. Because they have been described in detail by SC, here we just report the formulas needed later in this paper: • Near-side−far-side (NF) scattering theory1,4,5,18−41 using the (exact) Fuller decomposition45 of f(θR). NF theory is very useful for understanding interference structure in DCSs; it is used to analyze the DCS for the H + D2 reaction in sections 6 and 7. We decompose f(θR) into the sum of N and F subamplitudes

(pTH)

X = L, Q, SF, TH

(10)

J=0

σ(θR ) = |f (θR )|2

Clearly s̃SF(J) depends on just two parameters, N and JN. It has a single jump discontinuity (step) of value N. b. Top-Hat Parametrization (param TH and param THz). The top-hat parametrization is obtained from param SF by setting s̃SF(J) equal to 0 from J = 0 to J < JM where JM < JN. The three pieces are given by

SX̃ (J ) = s X̃ (J ) exp[iϕX̃ (J )]

∞

∑ (2J + 1)SJ̃ PJ(cos θR )

In eq 10, k is the initial translational wavenumber, J is the total angular momentum quantum number, S̃J is the Jth modified scattering matrix element, and PJ(•) is a Legendre polynomial of degree J. The dependence of f(θR), S̃J (and related quantities) on vi,ji→vf,jf has been omitted for notational simplicity from eq 10, and below, as has the label vi,ji from k. The corresponding DCS is given by

(pSF)

⎧ 0 for 0 ≤ J < J M ⎪ ⎪ N for J J JN ≤ ≤ s TH ̃ (J ) = ⎨ M ⎪ ⎪ 0 for J > J ⎩ N

1 2ik

⎤ 1⎡ 2i ⎢⎣PJ(cos θR ) ± Q J(cos θR )⎥⎦ 2 π

(9)

J = 0, 1, 2, ...

4. SCATTERING THEORY The theory presented in this section is more general than our application to the H + D2 reaction, because it also applies to the following generic state-to-state chemical reaction

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and QJ(•) is a Legendre function of the second kind of degree J. The corresponding N,F DCSs are given by (for θR ≠ 0, π) σ (N,F)(θR ) = |f (N,F) (θR )|2

A + BC(vi ,ji ,m i = 0) → AB(vf ,jf ,mf = 0) + C

where mi and mf are the initial and final helicity quantum numbers, respectively. The reaction is assumed to take place at a fixed total energy or, equivalently, at a fixed initial translational wavenumber. As in the H + D2 case, θR is used to

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We have also resummed1,20,21,23,25,34−37,39−41 the PWS in eq 10 up to three times (r = 3) before making the NF 11417



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Table 1. Values of the S Matrix Parameters for the Four Parametrizations, pL, pQ, pSF, and pTH, Together with pTHz, Which Is a Special Case of pTH with the Phase Set Equal to Zero ϕ̃ (J) = aJ2 + bJ +c

a

s̃(J)

parametrization

a

b

c

N

Jz

JN

other parameters

L Q SF TH THz

−0.0663a −0.0669b −0.0663a −0.0663a 0

π π π π 0

6.428 6.428 6.428 6.428 0

0.03 0.03 0.02243 0.02266 0.02266

28.62 34.21 25 25 25

19.2450c 19.0189d 25.e 25.e 25.e

d = −0.0032 q = 0.00013 JM = 21 JM = 21

Jg = b/(−2a) = 23.69. bJg = b/(−2a) = 23.48. ccalculated from eq 6. dcalculated from eq 7. eJN = Jz.

diverges as θR → 0°, it has the advantage that N and F semiclassical subamplitudes can be identified. • Classical semiclassical approximation (CSA). The CSA ignores the NF interference term in the PSA DCS. It is useful for understanding general trends in the DCS. It also diverges as θR → 0°.

decomposition. It is known that such resummations can be a very effective way of removing nonphysical structure from the unresummed (r = 0) N and F DCSs. The full f(θR) is independent of resummation. • Local angular momentum (LAM) theory. A LAM analysis provides information on the full and N,F local angular momenta that contribute to the scattering at a particular angle, under semiclassical conditions.1,19,21,23,35−41 The full LAM is defined by LAM(θR ) =

d arg f (θR ) dθR

5. VALUES OF THE S MATRIX PARAMETERS The values of the phase parameters used for pL, pQ, pSF, and pTH are reported in Table 1. We recall from section 3A that b has a fixed value of π. Also c has been assigned the value 6.428 for convenience. Table 1 also includes pTHz, which is a special case of pTH with the phase set equal to zero. The values of the remaining parameters in Table 1 were chosen to fit the small-angle DCS of the H + D2 reaction using numerical S matrix elements obtained from accurate quantum scattering calculations1,4,46 at a total energy of 2.00 eV for the Boothroyd−Keogh−Martin−Peterson potential energy surface.47 The corresponding translational energy is 1.81 eV, with the translational wavenumber being k = 26.4 Å−1. Notice we regard the DCS using the numerical S matrix elements to be the result from a computational experiment. It can be replaced by the true experimental DCS in due course. Table 1 shows that the values of a are the same for pL, pSF, pTH, and for pQ, a is only different by 0.0006. This small difference nevertheless has a significant effect on the DCS. For example, using a = −0.0663 for pQ, the dimensionless dif ferential cross section (dDCS), k2 σ(θR=0°), has the value 6.44 compared to k2 σ(θR=0°) = 6.80 for a = −0.0669.

(16)

and the N,F LAMs are obtained from LAM(N,F)(θR ) =

d arg f (N,F) (θR ) dθR

(17)

Resummation theory can also be applied to the PWS (10) before calculating the N,F LAMs; this helps the physical interpretation of the N,F results from the LAM analysis. The args in eqs 16 and 17 are not necessarily principal values. The full LAM is independent of resummation. Note that the full and N,F PWS of eqs 10−17 use as input SJ̃ for J = 0, 1, 2, ..., which is discontinuous everywhere. 4B. Semiclassical Glory Theory. A systematic semiclassical theory of forward glory scattering has been developed in refs 4 and 5. Applications can be found in refs 1, 4, 18, 19, 23, 38, and 41. Because the working equations have been presented in ref 4 and SC, here we just list the approximations used in section 7: • Integral transitional approximation (ITA). This is valid for angles on and close to the caustic direction, θR = 0°. It is a global approximation in that it requires information on S̃(J) for all values of J. • Uniform semiclassical approximation (USA). This is the most accurate semiclassical theory currently available; it is valid both on and away from θR = 0°. It is a local approximation because it assumes the forward glory scattering receives most of its contribution from two real stationary-phase points close to J = Jg, provided that the corresponding values of |S̃(J)| are not too small. • Primitive semiclassical approximation (PSA). This is obtained when the Bessel functions in the USA are replaced by their asymptotic forms. Although the PSA

6. PARTIAL WAVE RESULTS AND DISCUSSION This section is divided into four parts: section 6A compares the properties of the parametrized and numerical S matrix elements as J varies, and section 6B examines the corresponding DCSs. A NF analysis of the DCSs using the parametrized S matrix elements is reported in section 6C and a NF LAM analysis of the scattering is given in section 6D. 6A. Comparison of the Parametrized and Numerical S Matrix Elements. Figure 1 shows plots of |S̃(J)| and arg S̃(J) versus J for the L, Q, SF, and TH parametrizations with 0 ≤ J ≤ 40 (colored lines and curves). Also shown are the numerical S matrix data, which consist of 31 nonzero complex numbers (black solid circles) for 0 ≤ J ≤ 30. The numerical data have been continued to real values of J using cubic B-spline interpolation (black solid curves). For the interpolated data, we see that the |S̃(J)| curve in Figure 1a possesses two local maxima at J ≈ 15.2 and J ≈ 22.2, and two local minima at J ≈ 12.6 and J ≈ 18.5. None of the parametrized |S̃(J)| curves reproduces these extrema; in fact, there is no detailed agreement between the curves for the 11418



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Figure 2. Plot of Θ̃(J)/deg versus J. Black solid curve: numerical S matrix data. Red solid line: parametrizations L (a = −0.0663), SF, and TH, which have identical quantum deflection functions. Blue solid line: parametrization Q. Red dashed line: parametrization L (a = −π/ Jz). Also shown for parametrization L are the two values of the glory angular momentum variable, Jg, which satisfy Θ̃(J) = 0 for a = −0.0663 and a = −π/Jz. The lines for these two values of a end at J = Jz = 28.62; see eq pL.

solid lines). In effect, the parametrizations have smoothed the numerical data. The red dashed line in Figure 2 labeled “L (a = −π/Jz)” is discussed later in section 7B. 6B. Differential Cross Sections for the Parametrized and Numerical S Matrix Elements. Figure 3 compares the

Figure 1. (a) Black solid circles: |S̃J| versus J for the numerical S matrix data at integer values of J. Black solid curve: |S̃(J)| versus J, which is the continuation of {|SJ̃ |} to real values of J. Also shown is |S̃(J)| versus J for the parametrizations: L (red solid curve), Q (blue solid curve), SF (orange solid curve), and TH (green solid curve). The discontinuities in parametrizations SF and TH are indicated by orange and green vertical dashed lines, respectively. (b) Black solid circles: arg SJ̃ /rad versus J for the numerical S matrix data at integer values of J. Black solid curve: arg S̃(J)/rad versus J, which is the continuation of {arg SJ̃ / rad} to real values of J. Red solid curve: arg S̃(J)/rad versus J for the parametrizations L, SF, and TH. Blue solid curve: arg S̃(J)/rad versus J for parametrization Q. The red and blue solid curves have been extended to J = 40. Parametrizations L, SF, and TH have identical phases. Note that displacing the red and blue curves vertically (i.e., changing c in eq 4) is of no physical consequence, because the DCS is independent of c.

numerical and parametrized S matrix elements, although the decrease in |S̃L(J)| and |SQ̃ (J)| starts close to the second maximum and then drops in a way similar to that of the numerical data. In Figure 1b, we first note that displacing the parabolic ϕ̃ (J) curves vertically, which corresponds to changing c in eq 4, is of no physical significance, because the DCS is independent of c. We see that the ϕ̃ (J) parabolae have curvatures similar to that of the numerical data for the J region around the maxima of the ϕ̃ (J) curves, i.e., around the glory point: Jg = 23.7 for pL, pSF, pTH and Jg = 23.5 for pQ. It will be shown in the PWS and semiclassical analyses of sections 6B, 6C, and 7, respectively, that this J region is where the S matrix elements contribute most to the forward-angle DCS. The noticeable drop in arg S̃(J) on going from J = 18 to J = 19 is associated with the corresponding near-zero in |S̃(J)| visible in Figure 1a. For the semiclassical analysis of section 7, it is interesting to compare Θ̃(J) defined by eq 5 for the parametrized and numerical S matrices. This is done in Figure 2. It can be seen that the Θ̃(J) curve for the numerical data (black solid curve) possesses complicated oscillations, whereas for the parametrizations, the Θ̃(J) plots are straight lines (red and blue

Figure 3. Plot of PWS k2σ(θR) versus θR. Black curves: numerical S matrix data. Red curves: parametrization L. Blue curves: parametrization Q. Orange curves: parametrization SF. Green curves: parametrization TH. (a) 0° ≤ θR ≤ 30°, with the inset showing 0° ≤ θR ≤ 5°, and (b) 0° ≤ θR ≤ 180°.

dDCSs for the parametrized and numerical S matrix elements. In particular, Figure 3a shows the angular range, 0° ≤ θR ≤ 30°, with the inset displaying the dDCSs for 0° ≤ θR ≤ 5°. The dDCSs for the full angular range of 0° ≤ θR ≤ 180° are plotted in Figure 3b. 11419



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It can be seen that the parametrized and numerical dDCSs are almost indistinguishable up to about 10°, with good, although diminishing, agreement, as θR increases to 30°. For θR ≳ 30°, detailed agreement between the parametrized and numerical dDCS is lost. This is expected because it is known1,4,21−25,48 that the H + D2 reaction undergoes a change in mechanism for θR ≈ 45°. Nevertheless, the agreement between the parametrized and numerical dDCSs for θR ≲ 30° is surprising considering the discrepancies shown in Figure 1 for arg S̃(J), and especially |S̃(J)|. Especially surprising is the success of param SF and param TH in describing the forward scattering, because these parametrizations are usually considered too simplistic to describe DCSs. The success of pL, pQ, and pSF in Figure 3a is because of the close agreement of the curvatures of ϕ̃ (J) and the numerical arg S̃(J) for J ≈ Jg. The contribution from J values away from this stationary region is small because exp[iϕ̃ (J)] in eq 8 becomes rapidly oscillating. This can be demonstrated by replacing ϕ̃ (J) by a constant, e.g., zero, when it is found that the good agreement between the pL, pQ, pSF, and numerical dDCSs in Figure 3a is lost. An important exception is for pTH with zero phase, i.e., pTHz. Figure 4a shows there is good agreement between the dDCSs for pTHz and the numerical S matrix elements. To understand this unexpected result, we note that ϕ̃ TH(J) has the values 43.2, 43.5, 43.6, 43.6, 43.5 for J = 21, 22, 23, 24, 25, respectively. These values are almost constant, so it is a good approximation to replace ϕ̃ TH(J) by a constant in eq 8, or equivalently, by zero. [Also recall from Table 1 that s̃TH(J) = 0 for J outside of the window, J = 21(1)25]. The parametrization THz is the simplest we have found that can reproduce the scattering at forward angles. In fact, it is so simple that we can analytically sum the PWS and express f THz(θR) in closed form. It is proven in the Appendix that fTHz (θR ) =

N 1 {(J + 1)[PJ (x) − PJ + 1(x)] N N 2ik 1 − x N − JM [PJ

M

− 1(x)

− PJ (x)]} M

Figure 4. Angular scattering for parametrization THz. (a) Plot of PWS k2σ(θR) versus θR. Black curve: numerical S matrix data. Green curve: parametrization THz. The angular range is 0° ≤ θR ≤ 30°, with the inset showing 0° ≤ θR ≤ 5°. (b) NF PWS analysis for log(k2σ(θR)) versus θR for parametrization THz. Green curve: full PWS dDCS. Red curve: N r = 0 PWS dDCS. Blue curve: F r = 0 PWS dDCS. The red and blue curves are identical. (c) NF PWS analysis for LAM(θR) versus θR for parametrization THz. Green line: full PWS LAM, which is identically equal to zero. Red curve: N r = 0 PWS LAM. Blue curve: F r = 0 PWS LAM. The blue curve is the negative of the red curve.

(18)

where x = cos θR and x ≠ 1. We emphasize the surprising nature of our result for pTHz. An S matrix with five nonzero equal values, given by SJ̃ = 0.02266 for the window J = 21(1)25, can reproduce the angular scattering when θR ≲ 30° for the H + D2 reaction, as computed by a state-of-the-art quantum scattering calculation for a stateof-the-art potential energy surface. This unexpected result for pTHz suggests the following questions: If we restrict the PWS (10) for the numerical S matrix data to the window J = 21(1)25, will we see agreement with the PWS that sums over the full range of J values, i.e., from J = 0 to J = Jmax = 30? And, if we then make the replacement S̃J → |S̃J| in the J = 21(1)25 window, will we obtain agreement with the full PWS? These questions are examined in Figure 5, where we indeed observe quite good agreement for the numerical S matrix data between the dDCSs for the full and two window PWS calculations. In summary: Our results demonstrate that simple piecewisecontinuous S matrix elements of class C0 and piecewisediscontinuous S matrix elements of class C−1 can be used to model the forward scattering of the H + D2 reaction. In particular, window representations of the scattering amplitude can give us important insights into the range of J values that contribute to the reaction dynamics in PWS calculations.

6C. Near-Side−Far-Side Analyses of the Differential Cross Sections Using the Parametrized S Matrix Elements. The aim of this section is to understand in more detail the dDCSs produced by the parametrizations, in particular at large angles where the dDCSs are quite different. To do this, we have carried out N,F analyses of the PWS dDCSs and in section 6D we also report full and N,F LAMs, in both cases performing up to r = 3 resummations of the PWS (10). The N,F dDCS results are briefly described next. Figure 6 presents on a logarithmic scale the full, N and F dDCSs for pL, pQ, and pSF for no resummation (r = 0) of the PWS (10), as well as for three resummations (r = 3). The resummation parameters were determined as in SC. Now the Anni et al. procedure35 used by SC to calculate the resummation parameters assumes that the six leading S matrix 11420



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Figure 5. Angular scattering for numerical S matrix data. Plot of PWS k2σ(θR) versus θR. Black curve: PWS with partial waves, J = 0(1)30. Red curve: PWS restricted to the window, J = 21(1)25. Green curve: PWS restricted to the window, J = 21(1)25 and with S̃J → |S̃J|. The angular range is 0° ≤ θR ≤ 30°, with the inset showing 0° ≤ θR ≤ 5°.

elements, SJ̃ for J = 0(1)5, are all nonzero. Table 1 reveals that this is not the case for pTH and pTHz, so for these parametrizations we plot just the N,F r = 0 dDCSs. We also recall that the full PWS dDCS is independent of r. The diffraction (or high frequency) oscillations in the full dDCSs at forward angles in Figure 6a for pL and in Figure 6b for pQ damp out as θR increases. In contrast, diffraction oscillations are visible over the whole angular range for pSF in Figure 6c and are particularly pronounced for pTH and pTHz in Figures 6d and 4b, respectively. Next we consider the N,F dDCSs. For pL, pQ, and pSF, we see that resummation noticeably cleans the F r = 0 dDCSs, with unphysical oscillations being removed or rendered less apparent. The cleaning effect on the N r = 0 dDCSs is less pronounced. For all parametrizations in Figures 4b and 6, the forward-angle diffraction scattering arises from NF interference. At larger angles, the scattering is N dominant for pL and pQ. This is also true for pSF, with the residual oscillations being a NF interference effect (but note that the N dDCSs themselves possess low frequency oscillations). In contrast to pL, pQ, and pSF, the scattering for pTH at sideward angles is F dominant, then becomes N dominant as θR increases, before becoming F dominant again. At all angles, the pronounced high frequency oscillations arise from NF interference. A new N,F effect appears in Figure 4b for pTHz: The N and F dDCSs are equal resulting in pronounced NF oscillations across the whole angular range. To understand this result, note that S̃THz(J) = s̃TH(J) in eq 9 is real; hence, the summands in eq 13 are complex conjugates, which results in the identity, σ(N)(θR) = σ(F)(θR). This result can also be deduced the following closed-form expression for the N,F subamplitudes, which is derived in the Appendix (N,F) (θR ) = f THz

Figure 6. NF PWS analysis for log(k2σ(θR)) versus θR for (a) parametrization L, r = 0 (dashed curves) and r = 3 (solid curves), (b) parametrization Q, r = 0 (dashed curves) and r = 3 (solid curves), (c) parametrization SF, r = 0 (dashed curves) and r = 3 (solid curves), and (d) parametrization TH, r = 0 (dashed curves). In panels (a)−(d), red solid and dashed curves are for N PWS and blue solid and dashed curves are for F PWS.

N 1 {(J + 1)[Q J(N,F)(x) − Q J(N,F) (x)] +1 N N 2ik 1 − x N (x) − Q J(N,F)(x)]} − JM [Q J(N,F) −1 M

M

(19)

In eq 19, x = cos θR and x ≠ ±1. 6D. Near-Side−Far-Side and Full Local Angular Momenta Analyses Using the Parametrized S Matrix Elements. The N,F and full LAMs for pL, pQ, pSF, and pTH are displayed in Figure 7a−d, respectively. We see that the information contained in these plots is consistent with the information from the N,F analyses of the dDCSs in Figure 6.

For example, r = 3 resummation has helped clean the F r = 0 LAMs of unphysical oscillations for pL, pQ, and pSF, with a much smaller effect on the N r = 0 LAMs. The N LAMs for pL and pQ are similar to the N LAM for the repulsive collision of two hard spheres,1,4,19 and this is also 11421



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physically to a decaying wave that propagates around the reaction zone. Note: the standard Regge pole surface-wave interpretation27,28 does not apply here because none of the parametrizations is of class Cω. It is known that oscillations in N,F LAMs about zero are not physically significant.35,36 This behavior occurs at large angles in Figure 7, so in the following we concentrate on small angles, θR ∈ [2°, 50°]. It is useful to calculate ⟨LAM(F) r=3(θR)⟩, the mean F r = 3 LAM for θR ∈ [2°, 50°]. We obtain the values 26.1 and 27.1 for pL and pQ, respectively. Figure 1a shows these values correspond to the decreasing middle pieces of |SL̃ (J)| and |SQ̃ (J)|. For pSF, we find ⟨LAM(F) r=3(θR)⟩ = 25.0, which is the same as the position of the jump discontinuity. For pTH, we obtain ⟨LAM(F) r=0(θR)⟩ = 24.0, which is close to the location of the second jump discontinuity at JN = 25. Finally we consider pTHz. The behavior of the full and N,F ̃ (J) = s̃TH(J) in eq 9 LAMs in Figure 4c is different. Because STHz is real, it follows that f(θR) has a constant phase, so its phase derivative with respect to θR, which is the full LAM by eq 16, is identically zero. This can also be deduced from the closed-form expression in eq 18. Likewise for the N,F LAMs, it follows that LAM(N)(θR) ≡ −LAM(F)(θR), which can also be derived from the closed-form results in eq 19. At forward angles for pTHz, we find that ⟨LAM(F) r=0(θR)⟩ has a value of 23.7 for θR ∈ [2°, 50°], which lies between the jump discontinuities at JM = 21 and JN = 25.

7. SEMICLASSICAL RESULTS AND DISCUSSION In this section, we use the full PWS DCS and full PWS LAM results presented in section 6 to test the accuracy of the semiclassical glory theories outlined in section 4B. Then we compare the N,F DCSs and N,F LAMs from the semiclassical and PWS theories. This is an important comparison because there is no guarantee that the NF PWS decomposition (12)− (14) will produce physical meaningful N,F DCSs and N,F LAMs, even though it is mathematically exact.29,35,36 We report semiclassical results just for param L, because our findings for param Q are very similar. First we show results for the pL parameters in Table 1, where a = −0.0663. We will demonstrate that this value of a provides a test of the semiclassical theories for the full DCS and full LAM when θR ≲ 37°. Second we change the value of a to a = −π/Jz = −0.109769, which we show provides a test of the semiclassical theories for the full DCS and full LAM up to almost θR = 180°. Our calculations illustrate an advantage of the parametrized S matrix approach. It is easy to see how changes in a parametrized S matrix produce changes in the full and N,F DCSs and the full and N,F LAMs. This is much more difficult when the starting point is a potential energy surface. 7A. Parametrization L with a = −0.0663. Figure 8a compares on a logarithmic scale the uniform semiclassical approximation (USA) dDCS with the PWS dDCS for 0° ≤ θ R ≤ 180°. Also plotted is the integral transitional approximation (ITA) and the classical semiclassical approximation (CSA), which is useful because it shows general trends in the semiclassical results. The inset shows the dDCSs for PWS, USA, and CSA for the angular range, 17° ≤ θR ≤ 47°. It can be seen that the USA dDCS agrees well with the glory oscillations in the PWS dDCS at smaller angles, but for θR ≳ 37° agreement is suddenly lost and the USA dDCS becomes equal to the oscillation-free CSA dDCS. The primitive semiclassical approximation (PSA) dDCS (not shown) agrees closely with the USA dDCS except that it diverges as θR → 0°.

Figure 7. NF PWS analysis for LAM(θR) versus θR for (a) parametrization L, r = 0 (dashed curves) and r = 3 (solid curves), (b) parametrization Q, r = 0 (dashed curves) and r = 3 (solid curves), (c) parametrization SF, r = 0 (dashed curves) and r = 3 (solid curves), and (d) parametrization TH, r = 0 (dashed curves). In panels (a)−(d), red solid and dashed curves are for N PWS and blue solid and dashed curves are for F PWS.

the case for pSF when θR ≲ 50°. The N LAM for pTH exhibits a different behavior. It can be seen that the F LAMs are approximately constant, or slowly increasing, at forward angles, which corresponds 11422



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angles (except for θR ≈ 0°,180°). In contrast, the F PWS curve only agrees with the F PSA curve up to θR ≈ 30°. The F PSA log (dDCS) then tends to −∞ at θR = 37.4° and the F PSA dDCS equals zero for all θR ≥ 37.4°. To understand the above semiclassical results, we first examine the graph of Θ̃(J) versus J for pL in Figure 2. From eq 5, Θ̃(J) is a straight line given by Θ̃L(J ) = 2aJ + π J ∈ [0,Jz ]

(20)

where the restriction on J comes from the definition of the preexponential factor s̃L(J)in eq pL. Now Figure 2 shows that the N PSA subamplitude, where Θ̃(J) > 0, is defined for all θR∈(0,π), whereas the F PSA subamplitude, which has Θ̃(J) < 0, corresponds to J values in the range J ∈ (Jg, Jz) = (23.69, 28.62) or equivalently to angles, Θ̃ ∈ (Θ̃(Jg), Θ̃(Jz)) = (0°, −37.4°) using eq 20, or θR ∈ (0°, 37.4°). For J ≥ Jz, we have s̃L(J) ≡ 0, with the consequence that the F PSA DCS is identically zero for θR ≥ 37.4°. Figure 8c compares the N,F r = 3 PWS LAMs with the N,F PSA LAMs. Also plotted is the full PWS and USA LAMs. We see that the (full) USA LAM agrees well with the full PWS LAM at smaller angles but for θR ≳ 37° agreement is suddenly lost and the USA LAM becomes equal to the oscillation-free N PSA LAM, cf. Figure 8a for the dDCSs. The F PSA LAM is only plotted for θR < 37.4°. Explicit expressions are available for the F,N PSA LAMs, namely1 ⎡ b ± θR 1⎤ LAM(F,N) + ⎥ PSA (θR ) = ± ⎢ ⎣ ( −2a) 2⎦

which shows that both PSA LAMs increase linearly with θR and have the same slope; this behavior can be seen in Figure 8c. The N r = 3 PWS LAM is in good agreement with the N PSA LAM for all angles (except for θR ≈ 0°, 180°); both are similar to the N LAM for the collision of two hard spheres.1,4,19 The F r = 3 PWS LAM for θR < 37.4° is in fair agreement with the F PSA LAM. 7B. Parametrization L with a = −π/JZ = −0.109769. The tests of the semiclassical glory theories using a = −0.0663 in section 7A were mostly limited to θR ≲ 37°. In terms of the plot of Θ̃L(J) versus J in Figure 2 (red solid line), this limitation arises because Θ̃L(Jz) = −37.4°. Evidently, to test the semiclassical glory theories over the full angular range of 0° to 180°, we should change a so that Θ̃L(Jz) = −π. Solving eq 20 for this case, i.e., −π = 2aJz + π, gives a = −π/Jz = −0.109769. The corresponding graph of Θ̃L(J) versus J is drawn in Figure 2 as a red dashed line. It can be seen that both the N and F scattering now extend to θR ∈ (0°, 180°). We show in Figure 9 semiclassical and PWS plots for a = −π/Jz analogous to those in Figure 8 where a = −0.0663. There is good agreement between the USA and PWS dDCSs up to θR ≈ 170° in Figure 9a. The PSA dDCS (not shown) agrees closely with the USA dDCS except for θR → 0° where it diverges. The validity of the ITA now extends to θR ≈ 60° . The NF analysis for the dDCS in Figure 9b shows the reaction is F dominant until θR ≈ 130°. The N,F r = 3 PWS dDCSs are in good agreement with the N, F PSA dDCSs, respectively, up to about θR = 130°. Similar trends can be seen in the NF LAM analysis in Figure 9c. We conclude for the a = −π/Jz case that the USA is accurate, not only for forward and sideward scattering angles but also for angles close to the backward direction. Likewise, the N,F PSA

Figure 8. Semiclassical analysis of the angular scattering for parametrization L with a = −0.0663. (a) Plot of log(k2σ(θR)) versus θR. Black solid curve: PWS. Orange solid curve: USA. Light-blue solid curve: ITA for 0° ≤ θR ≤ 90°. Pink dashed curve: CSA. The inset shows log(k2σ(θR)) for the range 17° ≤ θR ≤ 47° for PWS, USA, and CSA. (b) NF analysis for log(k2σ(θR)) versus θR. Black solid curve: PWS. Purple solid curve: N PSA. Green solid curve: F PSA, which diverges at θR = 37.4°. Red dashed curve: N r = 3 PWS. Blue dashed curve: F r = 3 PWS. (c) NF analysis for LAM(θR) versus θR. Black solid curve: PWS. Orange solid curve: USA. Purple solid line: N PSA. Green solid line: F PSA, which ends at θR = 37.4°. Red dashed curve: N r = 3 PWS. Blue dashed curve: F r = 3 PWS.

The ITA dDCS agrees well with the PWS dDCS for θR ≲ 20°, although it incorrectly predicts that the dDCS possesses zeroes. On the axial caustic, we have k2 σ(θR=0°) = 6.76 for the PWS dDCS, and k2 σ(θR=0°) = 6.74 for the ITA dDCS, a difference of only 0.3%. A NF analysis is reported in Figure 8b, which compares on a logarithmic scale the N and F dDCSs for the PSA with the N,F r = 3 PWS dDCSs. The full PWS dDCS is also plotted. We see that both the N and F PSA subamplitudes contribute at small angles, so the glory oscillations arise (as expected) from NF interference. As θR increases, N scattering dominates and the N PWS dDCS is in good agreement with the N PSA dDCS at all 11423



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param Q could reproduce the angular scattering for θR ≲ 30°. Second we employed a piecewise-discontinuous pre-exponential factor and a quadratic phase. We found that both param SF and param TH could reproduce the forward-angle scattering, even though these class C−1 parametrizations are usually considered too simplistic for use in PWS calculations of DCSs. We discovered that the ultrasimplistic param THz, which has a phase of zero, can also reproduce the glory scattering at forward angles. The S matrix elements for param THz are real and consist of five nonzero equal values, given by SJ̃ = 0.02266, for the window J = 21(1)25. Param THz is sufficiently simple that we could derive closed forms for f(θR), f(N)(θR), and f(F)(θR). We found that window representations of the PWS f(θR) can give us important insights into the range of J values that contribute to the reaction dynamics. The angular scattering for the parametrizations is considerably different at larger angles. We were able to gain physical insight into the side- and large-angle structure in the DCSs by decomposing f(θR) into f(N)(θR) and f(F)(θR) PWS subamplitudes. We also carried out full and NF LAM analyses of the scattering dynamics. In addition, we demonstrated that making up to three resummations of the PWS f(θR), before making the NF decomposition, leads to an improved physical understanding of structure in the DCSs and LAMs. Finally, we investigated the accuracy of the ITA, USA, PSA, and CSA semiclassical glory theories for the DCS of param L. By varying one phase parameter for param L, we showed that the USA is accurate from θR = 0° to close to θR = 180°. More generally, the semiclassical theories show that one or two (nonintegral) values of J contribute to each scattering angle, which helps us understand why the simple PWS parametrizations developed in this paper successfully reproduce the DCS at forward angles. Our approach is an example of a “weak” form of Heisenberg’s S matrix program, which does not use a potential energy surface(s); rather it focuses on the properties of the S matrix. Our method is easy to apply to DCSs from experimental measurements or from computer simulations.

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APPENDIX In this Appendix, we derive closed-form expressions for the full PWS scattering amplitude and the N,F r = 0 PWS subamplitudes for parametrization THz.

Figure 9. Semiclassical analysis of the angular scattering for parametrization L with a = −π/Jz = −0.109769. (a) Plot of log(k2σ(θR)) versus θR. Black solid curve: PWS. Orange solid curve: USA. Light-blue solid curve: ITA. Pink dashed curve: CSA. (b) NF analysis for log(k2σ(θR)) versus θR. Black solid curve: PWS. Purple solid curve: N PSA. Green solid curve: F PSA. Red dashed curve: N r = 3 PWS. Blue dashed curve: F r = 3 PWS. (c) NF analysis for LAM(θR) versus θR. Black solid curve: PWS. Orange solid curve: USA. Purple solid line: N PSA. Green solid line: F PSA. Red dashed curve: N r = 3 PWS. Blue dashed curve: F r = 3 PWS.

A1. Full Scattering Amplitude

Equations 9 and 10 show that we have to sum analytically N 2ik

fTHz (θR ) =

J = JN

∑ (2J + 1)PJ(x)

x = cos θR

J = JM

(A.1)

dDCSs and N,F PSA LAMs show that the corresponding PWS quantities are physically meaningful over a wide range of angles.

Now if we write the finite sum as the difference J = JN

8. CONCLUSIONS In this paper, we asked the following question about the

J = JN

J = JM − 1

∑ =∑ − ∑ J = JM

H + D2 (vi=0,ji =0) → HD(vf =3,jf = 0) + D

J=0

(A.2)

J=0

we can exploit the following identity

reaction at a total energy of 2.00 eV. Can we replace the analytic functions used by SC with simpler mathematical functions and still reproduce the forward-angle glory scattering? Employing the same design principles as SC, we first constructed S matrix elements of class C0 from a quadratic phase and a piecewise-continuous pre-exponential factor consisting of three pieces. We found that both param L and

J

sP(x) ≡

∑ (2J′ + 1)PJ′(x) J ′= 0

=

(J + 1) [PJ(x) − PJ + 1(x)] 1−x

J = 0, 1, 2, ... (A.3)

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for x ≠ 1 which is used in the theory of window representations (e.g., see eqs B.4 and B.5 of ref 40). Applying the results A.2 and A.3 to f THz(θR), we find that the PWS in eq A.1 can be written in the closed form (x ≠ 1)

ACKNOWLEDGMENTS We thank Dr. Chengkui Xiahou for detailed comments. Support of this research by the U.K. Engineering and Physical Sciences Research Council is gratefully acknowledged.

■

N 1 {(J + 1)[PJ (x) − PJ + 1(x)] N N 2ik 1 − x N

fTHz (θR ) =

– JM [PJ

M

− 1(x)

− PJ (x)]} M

A2. Near-Side−Far-Side Scattering Subamplitudes

Next we derive closed forms for the N,F subamplitudes. Equations 13 and 14 show that we must also evaluate a sum over QJ′(x) analogous to the one for sP(x) in eq A.3. We have not found in the literature a closed form for J

∑ (2J′ + 1)Q J′(x)

J = 0, 1, 2, ... (A.4)

J ′= 0

so we derive one below. We start with Christoffel’s Second Summation Formula, which states49,50 n

(y − x) ∑ (2k + 1)Pk(y) Q k(x) k=0

= (n + 1)[Pn + 1(y) Q n(x) − Pn(y) Q n + 1(x)] − 1

(A.5)

In the identity (A.5), we put y = 1, use the fact that Pk(1) = 1 for all k, and then make the replacements k → J′, n → J to obtain the following closed form for the PWS (A.4) for x ≠ ±1. sQ (x) =

⎛ J + 1⎞ 1 ⎜ ⎟[Q (x) − Q (x)] − J+1 ⎝1 − x ⎠ J 1−x

(A.6)

Remark: It can be proven that the identity (A.5) is valid for −1 ≤ y ≤ 1 and “the cut” −1 < x < 1 (this is not obvious from refs 49 and 50, where the identity is quoted for x and y “off the cut”). Substituting the linear combination (14) into the PWS subamplitudes of eq 13 and applying eqs A.2, A.3, and A.6 into the resulting sums, we obtain the following closed forms for the N,F r = 0 PWS subamplitudes (N,F) (θR ) = f THz

N 1 {(J + 1)[Q J(N,F)(x) − Q J(N,F) (x)] +1 N N 2ik 1 − x N (x) − Q J(N,F)(x)]} − JM [Q J(N,F) −1 M

M

where x = cos θR and x ≠ ±1. The N,F DCSs are then given by eq 15.

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REFERENCES

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The full DCS is then given by eq 11.

sQ (x) ≡

Article

AUTHOR INFORMATION

Corresponding Author

*Electronic mail: [email protected]. Present Address †

The Department of Chemistry, University of Oxford, The Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ, England. Notes

The authors declare no competing financial interest. 11425



Article

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Application of Heisenberg's S Matrix Program to the Angular

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