Application of Heisenberg's S Matrix Program to the Angular

Article pubs.acs.org/JPCA

Terms of Use CC-BY

Application of Heisenberg’s S Matrix Program to the Angular Scattering of the State-to-State F + H2 Reaction Xiao Shan† and J. N. L. Connor* School of Chemistry, The University of Manchester, Manchester M13 9PL, United Kingdom ABSTRACT: This paper makes two applications of Heisenberg’s S matrix program (HSMP) to the differential cross section (DCS) of the benchmark reaction F + H2(vi = 0, ji = 0, mi = 0) → FH(vf = 3, jf = 3, mf = 0) + H, at a relative translational energy of 0.119 eV (total energy, 0.3872 eV), where v, j, m are vibrational, rotational, and helicity quantum numbers, respectively, for the initial and final states. (1) The first application employs a “weak” version of HSMP in which no potential energy surface (PES) is employed. It uses four simple S matrix parametrizations, two of which are piecewise continuous, and two are piecewise discontinuous, developed earlier by X. Shan and J. N. L. Connor (J. Phys. Chem. A 2012, 116, 11414−11426) for the state-to-state H + D2 reaction. We find that the small-angle DCS is reproduced for only θR ≲ 10° when compared with the DCS for a numerical S matrix obtained in a large-scale quantum scattering computation using a PES. Here θR is the reactive scattering angle. (2) In our second application, we ask the question “Can simple modifications to the parametrized S matrix be made in order to extend the agreement to larger angles?” To answer this question, we adopt a “hybrid” version of HSMP, as outlined by Shan and Connor (Phys. Chem. Chem. Phys. 2011, 13, 8392− 8406), which indirectly uses PES information. We make simple Gaussian-type modifications to both the modulus and argument of the S matrix. We then obtain agreement between the DCSs for the modified and numerical S matrices up to θR ≲ 70°, a significant improvement compared with θR ≲ 10° for the unmodified parametrizations. We find that modifying the argument but not the modulus, or modifying the modulus but not the argument, fails to extend the agreement to larger angles. A semiclassical analysis is used to prove that the enhanced small-angle scattering for the “modified-modulus−modified-argument” parametrized S matrix is an example of a forward glory.

1. INTRODUCTION

H + D2 (vi = 0, ji = 0, m i = 0)

1,2

This paper is the third in a series using a new approach for providing physical insight into the dynamics of state-to-state chemical reactions, in particular information contained in the angular scattering. Our new approach is inspired by Heisenberg’s scattering matrix programme (HSMP),3−6 which can be summarized by the scheme

→ HD(vf = 3, jf = 0, mf = 0) + D

where v, j, and m are vibrational, rotational, and helicity quantum numbers respectively, with the “i” and “f” subscripts denoting the initial and final states, respectively. At a relative translational energy of 1.81 eV (total energy, 2.00 eV), the DCS exhibits a pronounced forward peak, which arises from glory scattering,1,2,7−17 together with nearside−farside oscillations at larger angles1,2,8−16,18,19 (in both cases the angles are center-ofmass reactive scattering angles, denoted θR). SC1 showed that four simple S matrix parametrizations could accurately reproduce the structured small-angle peak for θR ≲ 30°, on comparison with a numerical DCS computed in a stateof-the-art scattering calculation using a state-of-the-art potential energy surface. This was an unexpected result. Each S matrix parametrization used a phase quadratic in J, the total angular momentum quantum number, together with a monotonically decreasing analytic function in J for the pre-exponential factor.

initial states → S matrix → final states

HSMP focuses on the properties of the S matrix because, in principle, it contains all the information needed to calculate observables, such as differential cross sections (DCSs), without the need for a potential energy surface(s). Heisenberg hoped to use general physical principles, e.g., unitarity, causality, and analyticity, to determine the S matrix. Unfortunately, Heisenberg’s hope has never been realized. We avoided this difficulty1,2 by using a “weak” form of HSMP (hereafter denoted wHSMP), in which four general physical principles relevant to state-to-state chemical reactions were employed to suggest simple parametrized forms for the S matrix, which then provided physical insight into structure in the DCSs. In our first paper (hereafter denoted SC1),1 we studied the fundamental reaction © 2014 American Chemical Society

Special Issue: Franco Gianturco Festschrift Received: March 31, 2014 Revised: May 19, 2014 Published: May 20, 2014 6560

dx.doi.org/10.1021/jp5031762 | J. Phys. Chem. A 2014, 118, 6560−6573

The Journal of Physical Chemistry A

Article

These four parametrizations are of class Cω in the notation used to classify the continuity and differentiability of functions.2 SC1 also included additional information and historical remarks on HSMP. In our second paper (hereafter denoted SC2),2 we again studied the H + D2 reaction using design principles for the S matrix that were the same as those used in SC1. But this time we asked the question “Can we replace the analytic functions in SC1 with simpler mathematical functions and still reproduce the small-angle scattering?” In particular, we used a quadratic phase and piecewise-continuous functions (of class C0) for the pre-exponential factor. We also employed piecewise-discontinuous functions (of class C −1 ) with one or two step discontinuities. A surprising finding2 was that these simple parametrizations could also reproduce the numerical DCS for θR ≲ 30°. In addition, we discovered2 an ultrasimplistic parametrization, in which the modified S matrix consisted of just five constant (nonzero) values, SJ̃ = 0.02266 for the window J = 21(1)25, could also reproduce the enhanced small-angle scattering for θR ≲ 30°. The purpose of this paper is to examine the angular scattering dynamics of a second state-to-state reaction, namely F + H2 → FH(vf = 3) + H, whose product DCS has been measured in the classic experiments of Neumark et al.20 using crossed molecular beams and more recently by Wang et al.21 Understanding the small-angle peak of the F + H2 reaction has also played a very important role in the theory of chemical reaction dynamics; for information on this, we rely on the review by Hu and Schatz.22 More generally, we note that the small-angle scattering of reactive collisions has been of interest for more than 40 years, with some early calculations described in refs 23−25. In this paper, we study the benchmark state-to-state transition

This encouraging result suggests another question: “Can a parametrized S matrix be found whose DCS agrees with the numerical DCS across the whole angular range, i.e., 0° ≤ θR ≤ 180°?” The answer to this question is yes. Such a parametrized S matrix has been constructed in ref 26 by one of us (J.N.L.C.). However, the construction of this parametrized S matrix required sophisticated techniques, such as Regge pole positions and residues, Padé approximants, the QP decomposition, etc. In contrast, the emphasis in this paper, as in SC1 and SC2, is the design of much simpler S matrix parametrizations which, nevertheless, still provide insights into the reaction dynamics. The general physical principles used in wHSMP for the design of parametrized S matrix elements are presented in section 2. We use the same four (simple) functional forms as SC2; they are described in section 3, together with their hHSMP modifications using Gaussian-type functions to reshape the modulus and argument. The scattering theories employed in this paper to calculate DCSs are outlined in section 4: we use both partial wave series (PWS) and semiclassical techniques. The parameter values for the unmodified S matrices are reported in section 5; the resulting PWS DCSs are then displayed and discussed in section 6. The values of the modification parameters are listed in section 7, and the corresponding PWS DCSs are plotted and examined in section 8. A semiclassical analysis in section 9 is used to prove that the enhanced small-angle scattering for the “modifiedmodulus−modified-argument” parametrized S matrix is an example of a forward glory. Our conclusions are in section 10. To provide additional physical insights into interference structure in the PWS DCSs, we also apply Nearside−Farside (NF) theory,1,2,18,19,27 including up to three resummations of the PWS.1,2,28−31 In addition, we report full and N,F local angular momentum (LAM) analyses1,2,28−32 of the angular scattering. Most of our PWS and semiclassical results are presented graphically.

F + H 2(vi = 0, ji = 0, m i = 0)

2. GENERAL PHYSICAL PRINCIPLES FOR PARAMETRIZED S MATRIX ELEMENTS In this section, we describe four general physical principles relevant to chemical reactions. Our design strategy for the construction of the S matrix elements is based on them. This approach is an example of wHSMP. As in SC1 and SC2, the following notation is used: we consider a (modified) S matrix ̃ ̃ element for a transition i → f and write Sf←i J , or more simply, SJ in which J = 0, 1, 2,..., is the total angular momentum quantum number. When SJ̃ is continued to real values of J, we write S̃(J). The following four general physical principles are those employed by SC2: (1) The forces responsible for chemical reactions are short ranged, of the order of 10−10 m. This implies S̃J → 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 the set {S̃J} to (a) a piecewise-continuous function, S̃(J) (of class C0), with simple properties for the pieces, e.g., the pre-exponential factor for each piece is a constant or is monotonically decreasing; or

→ FH(vf = 3, jf = 3, mf = 0) + H

at a relative translational energy of 0.119 eV (total energy, 0.3872 eV). Using design principles that are the same as those used in SC2, we find that four simple S matrix parametrizations of classes C0 and C−1, including the ultrasimplistic parametrization, S̃J = 0.07976 for the window J = 14, 15, 16, can again reproduce the small-angle peak when compared with a numerical DCS computed using quantum S matrix elements. We find these parametrized forms can accurately describe the small-angle scattering up to θR ≈ 10°; the DCSs from the parametrized forms then become less accurate as θR increases. Indeed, it is known9 that this F + H2 system has a change in mechanism at θR ≈ 43°. A natural question to ask about the F + H2 reaction is “Can simple modifications to the parametrized S matrix elements be made to extend the agreement to larger angles?” In SC1, a hybrid version of the wHSMP was outlined (hereafter denoted hHSMP) in which parametrized S matrix elements are refined by adding scattering information obtained from a potential energy surface(s). In this paper, we make simple Gaussian-type modifications to the modulus and argument of one of the parametrizations for the S matrix; the modifications are suggested by the properties of the numerical S matrix elements. We can then obtain agreement between the DCS from the modified S matrix and the numerical DCS up to θR ≲ 70°. 6561



Article

(b) a piecewise-discontinuous function, S(̃ J) (of class C−1), with simple properties, e.g., the pieces have constant preexponential 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. Notice that principle 3 was given in a stronger form in SC1, where it was assumed that S̃(J) is analytic, i.e., of class Cω, with a monotonically decreasing pre-exponential factor, whereas now S̃(J) can be of classes C0 or C−1. Keeping the above physical principles in mind, we follow SC2 and parametrize S̃(J) in the polar form S(̃ J ) = s (̃ J ) exp[iϕ(̃ J )]

from N to 0 either linearly or quadratically in J. For both parametrizations, s̃(J) depends on three independent parameters. 3A.b.1. Linear Parametrization (Param L or pL). This parametrization has three pieces defined 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 ⎩

From the properties of a straight line, it follows that JN = Jz + N /d = Jz − N /|d| with d < 0

Again, the parameters JN and Jz are nonintegral in general. Usually, the three parameters N, d, and Jz are specified first; then we calculate JN from eq 4. We see that |d| acts as a “diffuseness” parameter, which characterizes how quickly s̃(J) decreases from N to 0. The parameter Jz acts as a “cut-off” value of J. 3A.b.2. Quadratic Parametrization (Param Q or pQ). This parametrization also has three pieces given by

(1)

where ϕ̃ (J) is a real scattering phase (or argument) and s̃(J) is positive or zero, with 0 ≤ s̃(J) ≤ 1 and s̃(J) → 0 for J → ∞. Also we have, ϕ̃ (J) = arg S̃(J).

3. PARAMETRIZATIONS FOR THE S MATRIX ELEMENTS This section is divided into two parts. Section 3A defines unmodif ied parametrizations, which are four simple parametrizations for the S matrix that have the same functional forms as in SC2. Note that no potential energy surface is used in the construction of these S matrices. In section 3B, we define modified S matrix parametrizations, where the modifications to the modulus and argument are of a Gaussian type. 3A. Unmodified Parametrizations. 3A.a. Parametrization of ϕ̃ (J). As in SC1 and SC2, we use a quadratic phase in J for ϕ̃ (J), namely ϕ(̃ J ) ≡ ϕ(̃ J ; a , b , c) = aJ 2 + bJ + c

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

From the properties of a quadratic curve, we find

JN = Jz −

N /q

(5)

with q acting as a “diffuseness” parameter. Usually we calculate JN from eq 5, having first specified the three parameters N, q, and Jz. 3A.c. Piecewise-Discontinuous Parametrizations of s̃(J). We define two parametrizations for s̃(J) which are of class C−1 with one or two jump discontinuities (steps). 3A.c.1. Step-Function Parametrization (Param SF or pSF). This parametrization is obtained from pL or pQ in the limit Jz → JN. It consists of two pieces and is of a standard type. With JN = Jz, param SF is defined by

(2)

where a, b, and c are three real parameters with a < 0. It is convenient to include c even though its value does not influence the DCS. We recall the physical significance of a and b. First we define the quantum deflection f unction, Θ̃(J), by1,2,9,10 d argS(̃ J ) dϕ(̃ J ) Θ̃(J ) ≡ = = 2aJ + b dJ dJ

(4)

(3)

When J = 0, eq 3 shows that Θ̃(J = 0) = b; in practice, we adopt principle 4 in section 2 and set b = π in all our parametrizations so that a head-on collision corresponds to backward (or rebound) scattering of the products in a classical picture. An exception is the THz parametrization (defined below) which has ϕ̃ (J) ≡ 0. It also follows from eq 3 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); because Jg is positive, a must be negative (a < 0). 3A.b. Piecewise-Continuous Parametrizations of s̃(J). We parametrize the pre-exponential factor s̃(J) as a piecewisecontinuous function of J of class C0 (but not C1). In particular, the parametrizations have 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. The parameters JN and Jz are nonintegral in general. We use two simple parametrized forms for s̃(J), whereby it decreases

⎧ for 0 ≤ J ≤ JN ⎪N sSF ̃ (J ) = ⎨ ⎪ ⎩ 0 for J > JN

It is evident that s̃SF(J) depends on just two parameters, N and JN, and possesses a single jump discontinuity (step) of value N. 3A.c.2. Top-Hat Parametrization (Param TH or pTH). pTH can be obtained from pSF by setting s̃SF(J) equal to 0 from J = 0 to J < JM where JM < JN. The three pieces are given by ⎧ 0 for 0 ≤ J < J M ⎪ ⎪ N for J J JN ≤ ≤ ⎨ s TH ̃ (J ) = M ⎪ ⎪ 0 for J > J ⎩ N

Param TH depends on three parameters: N, JM, and JN, and possesses two jump discontinuities (steps), both of value N. Param TH is an example of a window representation for the S matrix,31,33,34 with the window in J-space being given by JM ≤ J ≤ JN. 3A.c.3. Top-Hat Parametrization with Zero Phase (Param THz or pTHz). This parametrization is obtained by setting the 6562



Article

phase of pTH equal to zero, ϕ̃ (J) ≡ 0. Param THz has the same window as pTH, namely, JM ≤ J ≤ JN. 3A.d. Full Unmodified S Matrix Elements. The full S matrix element is given by eq 1 for all parametrizations. As in SC2, we will sometimes attach the labels, L, Q, SF, and TH to S̃(J) and ϕ̃ (J) when we want to indicate which parametrization for s̃(J) is being used, i.e., SX̃ (J ) = s X̃ (J ) exp[iϕX̃ (J )]

X = L, Q, SF, TH

We refer to the parametrizations 12 and 13 as “param Marg” (or pMarg) and “param Mmod” (or pMmod) respectively. Notice that all three modified S matrices 11−13 belong to the same differentiability and continuity class as s̃(J). The Gaussian-type forms of eqs 8 and 10 have been chosen because Gaussian functions of J are often used in partial wave calculations to represent (part of) the S matrix.26,31,35−37 Also, the forms 8 and 10 allow the S matrix to be reshaped for small ranges of J, which are important when, for example, the dominance of the reaction changes from farside to nearside. Notice we use only our parametrizations for real values of J and do not consider the analytic properties of the pieces that make up S̃(J). The same approach we have used for DCSs can also be applied to integral cross sections, with the difference that only |S̃(J)| (or |S̃( J)|2) needs to parametrized. Any of the parametrizations in SC1, SC2, or in section 3 can be used for this purpose. If it is desired to exploit the correct analytical structure of the S matrix, then a more sophisticated approach is necessary.38

(6)

In addition, we have ̃ (J ) = s TH ̃ ( J )| STHz ̃ (J ) = |STH

In section 6C, we will use the subscript “U” to indicate an unmodified S matrix which has the same functional form as pQ but a different set of parameter values. 3B. Modified Parametrizations. We will find that S̃(J) obtained from quantum scattering computations is more structured than the simple parametrizations defined in section 3A. To partially allow for this complexity, we have used Gaussian-type functions to reshape the unmodified parametrizations. We will use the subscript “M” to indicate a modified S matrix. For the argument (or phase) of the modified S matrix, we write ̃ (J ) = ϕ(̃ J )g (J ) ϕMarg arg

4. SCATTERING THEORY The theory outlined in this section also applies to the following generic state-to-state chemical reaction A + BC(vi , ji , m i = 0) → AB(vf , jf , mf = 0) + C

(7)

where ϕ̃ (J) is the unmodified quadratic phase 2 and the modification factor is

The reaction is assumed to take place at a fixed total energy, or equivalently, at a fixed initial translational wavenumber. The reactive scattering angle θR is defined to be the angle between the incoming A atom and the outgoing AB molecule in the center-of-mass collision system. 4A. Partial Wave Theory. The scattering amplitude f(θR) is expanded in a basis set of Legendre polynomials. We write

garg (J ) ≡ garg (J ; a1 , J1 , d1; a 2 , J2 , d 2) =

1 + a1 exp( −(J − J1)2 /d1) 1 + a 2 exp( −(J − J2 )2 /d 2)

(8)

Note that after modification, the value of the parameter c in the original phase function, ϕ̃ (J), now does influence the DCS. For the modulus of the modified S matrix we have sMmod (J ) = s (̃ J )gmod (J ) ̃

f (θR ) =

gmod(J ) ≡ gmod(J ; a3 , J3 , d3; a4 , J4 , d4) 1 − a3 exp( −(J − J3)2 /d3) 1 + a4 exp( −(J − J4 )2 /d4)

(10)

(11)

(12)

f (θR ) = f (N) (θR ) + f (F) (θR )

and ̃ SMmod (J ) = sMmod (J ) exp[iϕ(̃ J )] ̃

(14)

(15)

The PWS, defined by eq 14, contains many numerically significant terms under semiclassical conditions, 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. Here we just report the formulas needed later because they have been described in detail by SC1. Nearside−Farside (NF) Scattering Theory1,2,18,19,27 Using the (Exact) Fuller Decomposition39 of f(θR). NF theory is very useful for understanding interference structure in DCSs. We decompose f(θR) into the sum of N and F subamplitudes

We refer to the parametrization 11 as “param Mmod/arg” (or pMmod/arg) for short. We will also employ partial modifications of the S matrix in which just ϕ̃ (J) is modified [but not s̃(J)] or s̃(J) is modified [but not ϕ̃ (J)]. We then write ̃ (J ) = s (̃ J ) exp[iϕ ̃ (J )] SMarg Marg

J=0

σ(θR ) = |f (θR )|2

The {ai,Ji,di} for i = 1, 2, 3, 4 are the modif ication parameters. Then the f ull modif ied S matrix is given by ̃ (J )] ̃ SMmod/arg (J ) = sMmod (J ) exp[iϕMarg ̃

∞

∑ (2J + 1)SJ̃ PJ(cos θR )

In eq 14, k is the initial translational wavenumber, J the total angular momentum quantum number, SJ̃ the Jth modified scattering matrix element, and PJ(•) a Legendre polynomial of degree J. The dependence of f(θR) and S̃J (and related quantities) on vi, ji, mi = 0 → vf, jf, mf = 0 has been omitted for notational simplicity from eq 14 and below, as has the label vi,ji from k. The DCS is then given by

(9)

where the unmodified s̃(J) is described in section 3A, and the modification factor is

=

1 2ik

(13)

(16)

with 6563



Article

Table 1. Values of the S Matrix Parameters for the Unmodified Parametrizations, pL, pQ, pSF, pTH, and pTHz ϕ̃ (J) = aJ2 +bJ + c

a

parametrization

a

L Q SF TH THz

−0.1013 −0.1013a −0.1013a −0.1013a 0

s̃(J)

b a

c

π π π π 0

N

3.435 3.435 3.435 3.435 0

0.075 79 0.075 79 0.067 10 0.079 76 0.079 76

JZ 18.0 19.74 16 16 16

JN

other parameters b

14.0 14.0c 16d 16d 16d

d = −0.01895 q = 0.0023 − JM = 14 JM = 14

Jg = b/(−2a) = 15.51. bCalculated from eq 4. cCalculated from eq 5. dJN = Jz.

f (N,F) (θR ) =

1 2ik

∞

briefly report the approximations needed for our application in section 9: • 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 Bessel Approximation (uBessel). 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 smallangle 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. Note that the uBessel approximation was called the uniform semiclassical approximation (USA) in SC1 and SC2. • Primitive Semiclassical Approximation (PSA). This is obtained when the Bessel functions in the uBessel approximation are replaced by their asymptotic forms. Although the PSA approximation diverges as θR → 0°, it has the advantage that N and F semiclassical subamplitudes can be identified. • Classical Semiclassical Approximation (CSA). The CSA approximation 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°.

∑ (2J + 1)SJ̃ Q J(N,F)(cos θR ) (17)

J=0

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

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

J = 0, 1, 2, ...

(18)

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

(19)

We say there is “nearside dominance” when σ (θR) > σ(F)(θR). If σ(N)(θR) is greater than σ(F)(θR) by a factor of 10 or less, there will typically be pronounced NF oscillations in σ(θR) arising from the interference of f(N)(θR) with f(F)(θR) in eq 16. The same is true for “farside dominance.”37 In addition, we have resummed1,2,28−31 the PWS in eq 14 up to three times (r = 3) before making the NF decomposition. Totenhofer et al.31 have presented a detailed account of resummation theory for a Legendre PWS. It is known that such resummations can be very effective for 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.1,2,28−32The full LAM is defined by 37

LAM(θR ) =

d argf (θR ) dθR

(N)

5. VALUES OF THE UNMODIFIED S MATRIX PARAMETERS The values of the phase parameters used for pL, pQ, pSF, and pTH are reported in Table 1. Notice that the values of a are the same for the four parametrizations. Also b has a fixed value of π (see the discussion in section 3A.a). For convenience, c has been assigned the value 3.435 (the DCS is independent of c). Table 1 also includes pTHz, whose phase is identically zero. The values of the remaining parameters in Table 1 were chosen to fit the small-angle DCS of the F + H2 reaction using numerical S matrix elements obtained from quantum scattering calculations9,40 at a total energy of 0.3872 eV for the Stark− Werner potential energy surface.41 The corresponding translational energy is 0.119 eV, with the translational wavenumber being k = 10.2 Å−1. As in SC1 and SC2, we regard the DCS obtained from the numerical S matrix elements to be the result from a computational experiment. In due course, it can be replaced by the true experimental DCS.

(20)

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

d argf (N,F) (θR ) dθR

(21)

We have again resummed the PWS (eq 14) up to three times before calculating the N,F LAMs; this helps the physical interpretation of the N,F results from the LAM analysis. The args in eqs 20 and 21 are not necessarily principal values. The full LAM is independent of resummation. When LAM(θR) < 0, we have nearside dominance; farside dominance corresponds to LAM(θR) > 0. 4B. Semiclassical Glory Theory. The semiclassical theory of small-angle glory scattering has been systematically developed in refs 9 and 10. Because the working equations have already been presented in ref 9 and SC1, here we just

6. PARTIAL WAVE RESULTS FOR THE UNMODIFIED S MATRICES This section is divided into two parts: section 6A compares the properties of the parametrized and numerical S matrix elements as J varies, and section 6B examines the corresponding DCSs. 6564



Article

6A. Comparison of the Unmodified and Numerical S Matrix Elements. Figure 1 shows plots of |S̃(J)| and arg S̃(J)

points, in theory, contribute to the semiclassical theory of glory scattering at small angles. However, upon examination of the corresponding numerical values of |S̃(J)|, we find that the contributions from J ≈ 17.8 and J ≈ 19.8 to the DCS are very small and can be neglected. Next we consider the parametrized curve ϕ̃ (J) = arg S̃(J), which is identical for pL, pQ, pSF, and pTH (see Table 1). Note that displacing the parabolic ϕ̃ (J) curve vertically, which corresponds to changing c in eq 2, is of no physical significance because the DCS is independent of c. We see in Figure 1b that the ϕ̃ (J) parabola has curvature similar to that of the numerical data for the J region around the maximum of the ϕ̃ (J) curve, i.e., around its glory point of Jg = 15.5. 6B. Differential Cross Sections for the Parametrized and Numerical S Matrix Elements. Figure 2 compares the

Figure 1. (a) Black solid circles: |S̃J| versus J for the numerical S matrix data at integer values of J. Black 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 curve), Q (blue 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 curve: arg S̃(J)/rad versus J, which is the continuation of {arg S̃J/rad} to real values of J. Red curve: arg S̃(J)/rad versus J for the parametrizations L, Q, SF, and TH, which have identical phases. Note that displacing the red curve vertically (i.e., changing c in eq 2) is of no physical consequence because the DCS is independent of c. Figure 2. Plot of PWS σ(θR) versus θR. Black curve: numerical S matrix data. Red curve: parametrization L. Blue curve: parametrization Q. Orange curve: parametrization SF. Green solid curve: parametrization TH. Dark-green dashed curve: parametrization THz. (a) 0° ≤ θR ≤ 40°, with the inset showing 0° ≤ θR ≤ 10°, and (b) 40° ≤ θR ≤ 180°.

versus J for the L, Q, SF, and TH parametrizations with 0 ≤ J ≤ 23 (colored lines and curves). Also shown are the numerical S matrix data {SJ̃ }, which consist of 24 nonzero complex numbers (black solid circles) for J = 0(1)23. The numerical data have been continued to real values of J using cubic B-spline interpolation (black solid curves); this is also the case for all the interpolations in this paper. For the interpolated data, we see that the |S̃(J)| curve in Figure 1a possesses two local maxima at J ≈ 9.4 and J ≈ 15.7 and a local minimum at J ≈ 13.0. None of the parametrized |S̃(J)| curves reproduce these extrema; in fact there is no detailed agreement between the curves for the numerical and parametrized S matrix elements, although the decrease in |S̃L(J)| and |S̃Q(J)| starts close to the second maximum and then drops in a way similar to the numerical data. Note that the steps in pSF and pTH occur in this same J range. Plots of arg S̃(J) versus J for the parametrized (red solid curve) and numerical S matrix elements, as well as the continuation of the numerical data, are shown in Figure 1b. We observe that the interpolated data possesses two maxima at J ≈ 15.6 and J ≈ 19.8, and a minimum at J ≈ 17.8. All these three

DCSs for the parametrized and numerical S matrix elements. In particular, Figure 2a shows the angular range 0° ≤ θR ≤ 40°; the inset displays the DCSs for 0° ≤ θR ≤ 10°. The DCSs for the angular range of 40° ≤ θR ≤ 180° are plotted in Figure 2b. It can be seen that the parametrized (colored curves) and numerical (black curve) DCSs possess oscillations across the full angular range. Up to about 10°, the DCS curves for the small-angle peak are almost indistinguishable. As θR increases to 40°, the amplitudes of the diffraction oscillations in the parametrized DCSs become too large, although their periods agree up to θR ≈ 20°. For θR ≳ 40°, agreement between the parametrized and numerical DCSs is lost, with the parametrized DCSs becoming too small for θR ≳ 75°. This behavior is expected because it is known9,40 that the F + H2 system undergoes a change in mechanism for θR ≈ 43°, from farside 6565



Article

Table 2. Values of the S Matrix Parameters for the Unmodified Parametrization, pU, and its Three Modifications, pMarg, pMmod, and pMmod/arg. The Values of the Modification Parameters, {ai, Ji, di}, for i = 1, 2, 3, 4, are given in Table 3 ϕ̃ (J) = aJ2 +bJ + c

s̃(J)

parametrization

a

b

c

garg

N

Jz

JN

gmod

Ua Marg Mmod Mmod/argd

−0.1013 −0.0849 −0.08 −0.0849

π π π π

3.435 −1.783 −1.783 −1.783

1b ≠1 1b ≠1

0.15 0.15 0.15 0.15

23.96 19.0 19.25 19.25

7.738c 13.87c 14.36c 14.36c

1 1 ≠1 ≠1

other parameters q q q q

= = = =

0.00057 0.0057, {ai,Ji,di} i = 1, 2 0.00628, {ai,Ji,di} i = 3, 4 0.00628, {ai,Ji,di} i = 1, 2, 3, 4

a pU has the same functional form as pQ. bThe DCS is independent of c for pU and pMmod. cCalculated from eq 5. dpMmod/arg is constructed from ϕ̃ Marg(J) and s̃Mmod(J).

and |S̃J| agree closely with each other over the whole angular range. 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 enhanced small-angle scattering of the F + H2 reaction. In addition, computational experiments reveal that 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. These results also hold for the H + D2 reaction1,2 and probably for many other reactions. 6C. Differential Cross Section for Parametrization U. In this section, we examine the DCS computed for a different set of parameter values for pQ as reported in Table 2. This will provide us with a clue how to modify the values of the parameters for pQ to get better agreement with the numerical DCS at larger angles (≳ 10°). To avoid confusion with the results already discussed in sections 6A and 6B, we use the notation “param U” (or pU) for the reparametrized Q in the following. Here U means “unmodified.” The values of the parameters listed in Table 2 show that ϕ̃ U(J) is the same as ϕ̃ Q(J), but s̃U(J) is different from s̃Q(J). In particular, the value of N is much larger than the previous value (0.15 versus 0.07579) and Figure 3 shows that the quadratic part of s̃U(J) (green curve) now extends over a range of J values much wider than that of s̃Q(J) (see Figure 1a).

dominant to nearside dominant; this is discussed later in the LAM analysis in sections 6C, 8A, 8B, and 8C. Despite this, the close agreement between the parametrized and numerical DCSs for θR ≲ 10° is surprising considering the discrepancies shown in Figure 1 for arg S̃(J), and especially for |S̃(J)|. Next we consider reasons for the success of the parametrized S matrix DCSs for θR ≲ 10°, exploiting our prior experience with the H + D2 reaction in SC1 and SC2. The success of pL, pQ, and pSF in Figure 2a is because of the close agreement of the curvatures of ϕ̃ (J) and the numerical arg S̃(J) for J ≈ Jg in Figure 1b. The contribution from J values away from this stationary region is small because exp [iϕ̃ (J)] in eq 6 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 DCSs in Figure 2a is lost. Next we consider the DCSs for pTH and pTHz, which are almost indistinguishable over the whole angular range. We also recall that pTH and pTHz are nonzero only for the window J = 14, 15, 16 (see Table 1) and that setting ϕ̃ TH(J) to zero gives us pTHz. The inset to Figure 2a shows there is good agreement for θR ≲ 10° between the DCS curves for the numerical S matrix, pTH, and pTHz. In order to understand these unexpected results, we note that ϕ̃ TH(J) has the values 27.56, 27.76, and 27.77 for the window J = 14, 15, and 16, respectively. These values are almost constant, so it is evidently a good approximation to replace ϕ̃ TH(J) by a constant in eq 6, or equivalently, by zero. The parametrization THz is the simplest we have found that can reproduce the scattering at small angles. In SC2 we analytically summed its PWS in a closed form, obtaining 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

(22)

where x = cos θR and x ≠ 1. The good agreement for the DCSs of pTH and pTHz with the numerical DCS at θR ≲ 10° suggests the following question: If we restrict the PWS (eq 14) for the numerical S matrix data to the window J = 14, 15, 16, will we see agreement with the PWS that sums over the full range of J values, i.e., from J = 0 to J = Jmax = 23? We have carried out this computational experiment and find that the two DCSs are in fair agreement for θR ≲ 20°, with the windowed DCS being systematically smaller. A follow-on question we can ask is If we make the replacement SJ̃ → |S̃J| in the J = 14, 15, 16 window, will we obtain agreement with the full PWS using the numerical S matrix data? Again we find that the two DCSs are in fair agreement for θR ≲ 20°. Also the two windowed DCSs using S̃J

Figure 3. Black solid circles: |SJ̃ | versus J for the numerical S matrix data at integer values of J. Black curve: |S̃(J)| versus J, which is the continuation of {|SJ̃ |} to real values of J. Green curve: |S̃(J)| versus J for parametrization U.

The DCS for pU (green curve) is shown in Figure 4. It can be seen there is still very close agreement with the numerical DCS (black curve) in Figure 4a for θR ≲ 10°. However, the scattering at larger angles for pU is now closer to the DCS for the numerical S matrix. Compare Figure 4b with Figure 2b. 6566



Article

Figure 4. Plot of PWS σ(θR) versus θR. Black curve: numerical S matrix data. Green curve: parametrization U. (a) 0° ≤ θR ≤ 40°, with the inset showing 0° ≤ θR ≤ 10°, and (b) 40° ≤θR ≤ 180°.

Figure 5. Full and NF r = 3 PWS analysis for LAM(θR) versus θR. (a) Numerical S matrix data. Black curve: full PWS LAM, and (b) parametrization U. Green curve: full PWS LAM. In both panels, red curves: N r = 3 PWS LAM. Blue curves: F r = 3 PWS LAM.

7. VALUES OF THE MODIFIED S MATRIX PARAMETERS The values of the S matrix parameters used for the unmodified parametrization, pU, and its modifications, pMarg, pMmod, and pMmod/arg, are reported in Table 2. Param Mmod/arg is constructed from param Marg and param Mmod as explained in section 8C. The values of the modification parameters {ai, Ji, di}, for i = 1, 2, 3, 4, which enter the modification factors, garg(J) and gmod(J), are given in Table 3. Notice that all three modified

Further insight into the inaccuracies of the pU DCS compared with the numerical DCS can be obtained with the help of a LAM analysis of the two PWS scattering amplitudes. Figure 5a shows plots of the full numerical LAM(θR) versus θR (black curve) together with its N and F r = 3 LAMs (red and blue curves respectively). It can be seen that the full LAM is positive up to θR ≈ 43° and then becomes negative as θR increases. The reaction is therefore farside dominant in the small-angle region but switches to nearside dominance for θR ≳ 43°. Analogous plots for the full (green curve) and N, F r = 3 LAMs (red and blue curves respectively) for pU are shown in Figure 5b. We observe that the full LAM stays negative at all angles, which indicates that the reaction is nearside dominant at all angles. This property of nearside dominance cannot be overcome by making small changes in the values of the parameters because the values of the parameters for pU already provide the best fit to the numerical DCS at small angles. To improve our pU parametrization, we are therefore left with two choices: develop a new model or make modifications to the existing model. In this paper, we do the latter. In section 8, we describe our results for three modified ̃ (J), and parametrizations of pU. We compare S̃Marg(J), SMmod ̃SMmod/arg(J) and their DCSs with the corresponding results for the numerical S matrix elements, where we also recall that pMarg, pMmod and pMmod/arg refer to eqs 12, 13, and 11, respectively. We shall also demonstrate that modifications to both the argument and modulus of SŨ (J) are necessary to describe the change in mechanism at θR ≈ 43°.

Table 3. Values of the Modification Parameters,{ai, Ji, di}, for i = 1, 2, 3, 4 g factor

parameter set i =

ai

Ji

di

garg

1 2 3 4

0.065 0.040 0.086 4.34

15.0 17.5 15.7 13.0

10.0 4.0 3.0 1.7

gmod

parametrizations have the same values of N, b(= π), and c(= −1.783), with the values of a and Jz being almost the same. Also, the three modified parametrizations are of class C0 because s̃U(J) is of class C0. Because we are indirectly using potential energy surface information in the construction of the parametrized S matrix, this is an example of hHSMP.

8. PARTIAL WAVE RESULTS FOR THE MODIFIED S MATRICES In this section we examine the effect on the DCSs and LAMs of modifying the argument of SŨ (J) (section 8A), then its modulus (section 8B), and finally modifying both the argument 6567



Article

̃ (J) because the modification is applied to the whole phase SMarg (see eq 7). Figure 7 compares the DCSs for pMarg (yellow curve) and the numerical data (black curve). It can be seen in Figure 7a

and modulus (section 8C). We also present the results of a N,F r = 3 analysis of the full DCSs and LAMs when this is useful for understanding the dynamics of the reaction. 8A. Modifying the Argument (Param Marg or pMarg). We first modify arg S̃U(J) according to eqs 7 and 8 to obtain a better fit to the numerical data in the glory range of J = 14 to J = 16, at the same time leaving the functional form of s̃U(J) unchanged. The parameters in s̃U(J) are then found by fitting the parametrized DCS to the numerical DCS. Tables 2 and 3 report the parameter values that we obtain, N.B., b has a fixed value of π. Figure 6 shows graphs of |S̃Marg(J)| and arg S̃Marg(J) versus J (yellow curves), where they are compared with the

Figure 7. Plot of PWS σ(θR) versus θR. Black curve: numerical S matrix data. Yellow curve: parametrization Marg. (a) 0° ≤ θR ≤ 40°, with the inset showing 0° ≤ θR ≤ 10°, and (b) 40° ≤θR ≤ 180°.

there is good agreement with the numerical DCS at small angles, in particular for 0° ≤ θR ≲ 10°. However, as θR increases, the pMarg DCS becomes less accurate. In particular, for 60° ≲ θR ≲ 120°, there is a large increase in intensity in the pMarg DCS, followed by better agreement at larger angles. In Figure 8, we plot LAM(θR) versus θR for pMarg (yellow curve) and the numerical data (black curve). We observe that the pMarg LAM is nearside dominated at all angles, unlike the numerical LAM. This result, together with the lack of agreement in the DCSs at intermediate angles, demonstrates

Figure 6. (a) Black solid circles: |S̃J| versus J for the numerical S matrix data at integer values of J. Black curve: |S̃(J)| versus J, which is the continuation of {|SJ̃ |} to real values of J. Yellow curve: |S̃(J)| versus J for parametrization Marg. (b) Black solid circles: arg SJ̃ /rad versus J for the numerical S matrix data at integer values of J. Black curve: arg S̃(J)/rad versus J, which is the continuation of {arg S̃J/rad} to real values of J. Yellow curve: arg S̃(J)/rad versus J for parametrization Marg.

corresponding plots for the numerical S matrix data (black solid circles and curves). We observe in Figure 6a for the moduli that the maximum at J ≈ 9.4 and the minimum at J ≈ 13.0 possessed by the interpolated data are not reproduced by pMarg. There is good agreement between the two curves only for J ≥ 16. Next we consider the arguments plotted in Figure 6b, where we see that the agreement is much better, especially in comparison with Figure 1b. We find that arg S̃Marg(J) has two maxima at J ≈ 15.5 and J ≈ 19.6 together with a minimum at J ≈ 18.3. These values are close to those for the interpolated data, namely J ≈ 15.6, J ≈ 19.8, and J ≈ 17.8, respectively. We also observe that arg S̃Marg(J) is in very good agreement with the interpolated data for J ≤ 16. Finally, we recall from section 3B that the parameter c cannot be chosen arbitrarily in arg

Figure 8. Plot of full PWS LAM(θR) versus θR. Black curve: numerical S matrix data. Yellow curve: parametrization Marg. 6568



Article

which is the region that contributes to the forward glory for the numerical data, we find agreement for the pMmod DCS is much worse when compared with the numerical DCS. Figure 10 compares the DCSs for the parametrized (pink curve) and numerical data (black curve). It can be seen that the

that our argument modification has failed to explain the mechanistic details of the F + H2 reaction. 8B. Modifying the Modulus (Param Mmod or pMmod). We now present results for a different modification of SŨ (J), which is constructed by multiplying s̃U(J) by gmod(J) ( see eqs 9 and 10) to obtain closer agreement with the numerical data. The functional form of arg SŨ (J) is left unchanged, i.e., as a quadratic function of J. We then adjust the parameter a in ϕ̃ U(J) to provide the best fit between the parametrized and numerical DCSs. The values of the parameters are reported in Tables 2 and 3. Note that b has a fixed value of π and the DCS is independent of the value of c for this modification. Figure 9 plots |S̃Mmod(J)| and arg S̃Mmod(J) versus J (pink curves). It can be seen that the curve for |S̃Mmod(J)| in Figure 9a

Figure 10. Plot of PWS σ(θR) versus θR. Black curve: numerical S matrix data. Pink curve: parametrization Mmod. (a) 0° ≤ θR ≤ 40°, with the inset showing 0° ≤ θR ≤ 10°, and (b) 40° ≤ θR ≤ 180°.

agreement at θR ≲ 20° is generally less accurate than that of either pU or pMarg. The pMmod and numerical DCSs are then in better agreement up to θR ≈ 45°, after which the two curves separate again. Graphs of LAM(θR) versus θR for pMmod and the numerical data are plotted in Figure 11 (pink and black curves, respectively). We see that the reaction is again nearside dominated for pMmod at all angles. We therefore conclude that our modulus modification has again failed to explain the mechanistic details of the F + H2 reaction.

Figure 9. (a) Black solid circles: |S̃J| versus J for the numerical S matrix data at integer values of J. Black curve: |S̃(J)| versus J, which is the continuation of {|SJ̃ |} to real values of J. Pink curve: |S̃(J)| versus J for parametrization Mmod. (b) Black solid circles: arg SJ̃ /rad versus J for the numerical S matrix data at integer values of J. Black curve: arg S̃(J)/rad versus J, which is the continuation of {arg S̃J/rad} to real values of J. Pink curve: arg S̃(J)/rad versus J for parametrization Mmod. Note that displacing the pink curve vertically (i.e., changing c in eq 2) is of no physical consequence because the DCS is independent of c.

is in good agreement with the corresponding numerical black curve for J ≳ 10. In particular, |S̃Mmod(J)| possesses a minimum at J ≈ 13.0 and a maximum at J ≈ 15.2, which are close to the values for the interpolated curve, J ≈ 13.0 and J ≈ 15.7, respectively. Figure 9b shows the corresponding graphs for the ̃ arguments. We see that arg SMmod (J) has a parabolic shape with a maximum at J = Jg = b/(−2a) = 19.635, which is close to the second maximum in the numerical argument plot (black curve). It was mentioned in section 6A that the semiclassical contribution to the DCS from this glory point is very small. ̃ However, if we change the maximum of arg SMmod (J) to ≈15.6,

Figure 11. Plot of full PWS LAM(θR) versus θR. Black curve: numerical S matrix data. Pink curve: parametrization Mmod. 6569



Article

8C. Modifying the Modulus and Modifying the Argument (Param Mmod/arg or pMmod/arg). We now consider the case where both the modulus and argument of pU are modified. In particular, the modifications already used for pMarg and pMmod in sections 8A and 8B, respectively, are ̃ employed, i.e., we construct SMmod/arg (J) from ϕ̃ Marg(J) and s̃Mmod(J). The values of the parameters are listed in Tables 2 and 3. In panels a and b of Figure 12 we present plots of the

Figure 13. Plot of PWS σ(θR) versus θR. Black curve: numerical S matrix data. Orange curve: parametrization Mmod/arg. (a) 0° ≤ θR ≤ 40°, with the inset showing 0° ≤ θR ≤ 10°, and (b) 40° ≤ θR ≤ 180°.

of pU can extend the region of agreement from θR ≲ 10° to θR ≲ 70°. At larger angles, θR ≳ 70°, Figure 12 shows this good agreement is lost, as was anticipated in the previous paragraph. The full LAMs for pMmod/arg and the numerical data are compared in Figure 14 (orange and black curves, respectively). It can be seen that both LAMs are positive at small angles and become negative at larger angles. There is good agreement between the two LAMs up to θR ≈ 100°. The pMmod/arg LAM predicts a change in reaction mechanism at θR ≈ 45°, which is close to the value θR ≈ 43° from the numerical S matrix data. A NF analysis of the full DCS and full LAM for pMmod/arg is presented in panels a and b of Figure 15, respectively. The full curves are plotted orange, the N curves red, and the F curves blue. Results are presented for no resummation (r = 0) of the PWS (eq 14) and for three resummations (r = 3) before carrying out the NF decomposition. For both the N,F DCSs and N,F LAMs, we observe that unphysical oscillations are removed or displaced to larger angles, i.e., “cleaning” has occurred, resulting in smoother r = 3 N,F curves. This cleaning effect is most evident for the F case. First we examine Figure 15a. Clearly visible in the N,F DCSs is the change in mechanism from F dominance at small angles to N dominance at larger angles. The diffraction oscillations arise from NF interference. These findings are confirmed in Figure 15b for the LAMs. For θR ≳ 45°, both the r = 0 and r = 3 N LAMs are seen to increase (but decrease in magnitude) approximately monotonically. This is the behavior expected for a reaction dominated by repulsive interactions of the hardsphere type.30 In contrast, the r = 3 F LAM is approximately constant for 5° ≲ θR ≲ 110°, with a mean value of 16.4, which Figure 12a shows corresponds to the high J side of the peak in the graph of |S̃Mmod/arg(J)| versus J. Physically, the result

Figure 12. (a) Black solid circles: |S̃J| versus J for the numerical S matrix data at integer values of J. Black curve: |S̃(J)| versus J, which is the continuation of {|SJ̃ |} to real values of J. Orange curve: |S̃(J)| versus J for parametrization Mmod/arg. (b) Black solid circles: arg SJ̃ /rad versus J for the numerical S matrix data at integer values of J. Black curve: arg S̃(J)/rad versus J, which is the continuation of {arg SJ̃ /rad} to real values of J. Orange curve: arg S̃(J)/rad versus J for parametrization Mmod/arg.

modulus and argument of the pMmod/arg S matrix elements versus J, respectively (orange curves), together with the corresponding graphs for the numerical data (black solid circles and curve). Note that these two plots are the same as Figures 9a and 6b, respectively. They are shown together to emphasize that simple Gaussian-type modifications to both the modulus and argument of S̃U(J) brings S̃Mmod/arg(J) into much better agreement with the numerical S matrix elements for the important glory region around J = 15. The main discrepancies in Figure 12 occur for |S̃Mmod/arg(J)| at low partial waves, where J ≤ 10. We anticipate that these discrepancies at low J will be revealed at large angles in the DCS. We compare the DCSs for the numerical and parametrized cases in Figure 13 (black and orange curves, respectively). It can be seen there is good agreement in the small-angle region; moreover, this good agreement extends to θR ≈ 70°. This agreement is clearly better than that achieved by the unmodified pU or the modified pMmod and pMarg. Thus, we have demonstrated that simple Gaussian-type modifications 6570



Article

LAMF(θR) ≈ constant corresponds to a decaying wave propagating around the reaction zone.14,30,40 The N,F results for pMmod/arg described in this section are similar to those for the numerical S matrix data, as previously discussed in detail by one of us (J.N.L.C.) and Anni.30

9. SEMICLASSICAL GLORY ANALYSIS FOR PARAMETRIZATION MMOD/ARG In this section, we present a semiclassical glory analysis for param Mmod/arg. The key quantity in the semiclassical analysis9−16 is the quantum def lection f unction, already met in eq 3. In the present case, we have ̃ (J ) ̃ dϕMarg d argSMmod/arg (J ) Θ̃Mmod/arg (J ) = = dJ dJ Figure 16 plots Θ̃(J) versus J for the numerical data (black curve) and for pMmod/arg (orange curve). It can be seen that

Figure 14. Plot of full PWS LAM(θR) versus θR. Black curve: numerical S matrix data. Orange curve: parametrization Mmod/arg. (a) 0° ≤ θR ≤ 180° and (b) 0° ≤ θR ≤ 50°.

Figure 16. Plot of Θ̃(J)/deg versus J. Black curve: numerical S matrix data. Orange curve: parametrization Mmod/arg. The glory angular momentum variable, Jg, satisfies Θ̃(Jg) = 0.

the two curves have the same general shape, with close agreement for J values around the first glory point at J = Jg. For the numerical data, we have Jg ≈ 15.55 and for pMod/arg, Jg ≈ 15.45. This result helps explain the good agreement between the two DCSs in Figure 13 for θR ≲ 70°. Also notice that a linear approximation for Θ̃(J) for pMmod/arg in Figure 16 is very accurate for J ≈ Jg. We have previously discussed in section 8C that there is a change in mechanism at θR ≈ 43°. This means we can apply only the semiclassical glory theory outlined in section 4B for θR ≲ 40°. Our results for the ITA, uBessel approximation, and CSA DCSs (colored curves) are compared with the PWS DCS (black curve) for the range 0° ≤ θR ≤ 40° in Figure 17. For clarity, both linear and log plots are shown. It can be seen that the ITA DCS agrees closely with the PWS DCS for θR ≲ 10°; at larger angles, it becomes bigger than the PWS DCS at the diffraction maxima, but smaller at the diffraction minima. The uBessel approximation DCS is in good agreement with the PWS DCS for θR ≲ 30°; then it becomes less accurate in both amplitude and period. The CSA curve, as expected, shows monotonic behavior; it is useful for understanding the general trend in the angular scattering. In summary, the semiclassical DCSs in Figure 17 prove that the scattering at small angles is an example of a forward glory for pMmod/arg. This conclusion is consistent with that for the numerical S matrix data, as analyzed in ref 9.

Figure 15. Full and NF PWS analysis for parametrization Mmod/arg. (a) log σ(θR) versus θR and (b) LAM(θR) versus θR. In both panels, orange curves: full PWS. Red dashed curves: N r = 0 PWS. Red solid curves: N r = 3 PWS. Blue dashed curves: F r = 0 PWS. Blue solid curves: F r = 3 PWS.

6571



Article

to extend the agreement to larger angles. We reported the results of a semiclassical DCS analysis for pMmod/arg, which proved that the scattering at small angles is an example of a forward glory.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address †

X.S.: The Department of Chemistry, University of Oxford, The Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, U.K. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Dr. Chengkui Xiahou for detailed comments. Support of this research by the UK Engineering and Physical Sciences Research Council and a UK Leverhulme Trust Emeritus Fellowship to J.N.L.C. is gratefully acknowledged.

■

REFERENCES

(1) Shan, X.; Connor, J. N. L. Angular Scattering using Parameterized S Matrix Elements for the H + D2(vi = 0, ji = 0) → HD(vf = 3, jf = 0) + D Reaction: An Example of Heisenberg’s S Matrix Programme. Phys. Chem. Chem. Phys. 2011, 13, 8392−8406. (2) Shan, X.; Connor, J. N. L. 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. J. Phys. Chem. A 2012, 116, 11414−11426. (3) Heisenberg, W. Die “beobachtbaren Gröβen” in der Theorie der Elementarteilchen. Zeit. Phys. 1943, 120, 513−538. Part I. Reprinted in Werner Heisenberg, Gesammelte Werke/Collected Works, Series A, Part II Original Scientific Papers/Wissenschaftliche Originalarbeiten; Blum, W., Dürr, H.-P., Rechenberg, H., Eds.; Springer-Verlag: Berlin, Germany, 1989; pp 611−636. (4) Heisenberg, W. Die beobachtbaren Gröβen in der Theorie der Elementarteilchen. II. Zeit. Phys. 1943, 120, 673−702. Part II. Reprinted in Werner Heisenberg, Gesammelte Werke/Collected Works, Series A, Part II Original Scientific Papers/Wissenschaftliche Originalarbeiten; Blum, W., Dürr, H.-P., Rechenberg, H., Eds.; SpringerVerlag: Berlin, Germany, 1989; pp 637−666. (5) Heisenberg, W. Die beobachtbaren Gröβen in der Theorie der Elementarteilchen. III. Zeit. Phys. 1944, 123, 93−112. Part III. Reprinted in Werner Heisenberg, Gesammelte Werke/Collected Works, Series A, Part II Original Scientific Papers/Wissenschaftliche Originalarbeiten; Blum, W., Dürr, H.-P., Rechenberg, H., Eds.; SpringerVerlag: Berlin, Germany, 1989; pp 667−686. (6) Heisenberg, W. Die Behandlung von Mehrkörperproblemen mit Hilfe der η-Matrix [Die beobachtbaren Gröβen in der Theorie der Elementarteilchen. IV]. Unpublished paper. Part IV. Reprinted in Werner Heisenberg Gesammelte Werke/Collected Works, Series A, Part II Original Scientif ic Papers/Wissenschaf tliche Originalarbeiten. Blum, W., Dürr, H.-P., Rechenberg, H., Eds.; Springer-Verlag: Berlin, Germany, 1989; pp 687−698 (original manuscript dated 1944 by the editors). (7) Adam, J. A. The Mathematical Physics of Rainbows and Glories. Phys. Rep. 2002, 356, 229−365. (8) Sokolovski, D. S. Glory and Thresholds Effects in H + D2 Reactive Angular Scattering. Chem. Phys. Lett. 2003, 370, 805−812. (9) Connor, J. N. L. Theory of Forward Glory Scattering for Chemical Reactions. Phys. Chem. Chem. Phys. 2004, 6, 377−390. (10) Connor, J. N. L. Theory of Forward Glory Scattering for Chemical Reactions: New Derivation of a Uniform Semiclassical Formula for the Scattering Amplitude. Mol. Phys. 2005, 103, 1715− 1725.

Figure 17. Semiclassical analysis of the small-angle scattering for parametrization Mmod/arg. (a) Plot of σ(θR) versus θR and (b) log σ(θR) versus θR. In both panels, 0° ≤ θR ≤ 40°. Black curve: PWS. Red curve: uBessel approximation. Blue curve: ITA. Orange dashed curve: CSA.

10. CONCLUSIONS SC1 and SC2 applied wHSMP to the state-to-state H + D2 reaction and showed that simple parametrizations of the S matrix could reproduce the angular scattering for θR ≲ 30°. This was an unexpected result; it was not thought that such simple parametrizations could reproduce the DCS for even a limited range of angles. The present paper applied the same four functional forms as used by SC2 to another benchmark state-to-state reaction, F + H2. We found that the small-angle DCS could again be reproduced when compared with the DCS from a large-scale quantum PWS scattering computation using a quantum potential energy surface. The simplest parametrization we discovered was pTHz, which consists of three nonzero equal values, given by S̃J = 0.07976, for the window J = 14, 15, 16. However, the agreement for all the functional forms we investigated extended to only θR ≲ 10°. We then asked the question, “Can simple modifications be made to the parametrized S matrix in order to extend the agreement to larger angles?” In particular, we used hHSMP in which parametrized S matrix elements are modified by adding scattering information suggested by general trends in the numerical S matrix elements. In our application to pU, we used simple Gaussian-type modifications to both the modulus and argument of the S matrix. We then obtained agreement between the DCS for pMmod/arg and the numerical DCS up to θR ≲ 70°, a significant improvement compared with θR ≲ 10° for the unmodified parametrizations. We found that modifying the argument but not the modulus (pMarg), or modifying the modulus but not the argument (pMmod), failed 6572



Article

(11) Xiahou, C.; Connor, J. N. L. Accuracy of a Uniform Semiclassical Approximation for the Forward Glory Scattering of Chemical Reactions. In Semiclassical and Other Methods for Understanding Molecular Collisions and Chemical Reactions; Sen, S., Sokolovski, D., Connor, J. N. L., Eds.; Collaborative Computational Project on Molecular Quantum Dynamics (CCP6); Daresbury Laboratory: Warrington, U.K., 2005; pp 44−49. (12) Xiahou, C.; Connor, J. N. L. Theory of Forward Glory Scattering for Chemical Reactions: Accuracy of Semiclassical Approximations using a J-shifted Eckart Parameterization for the Scattering Matrix Element. Mol. Phys. 2006, 104, 159−175. (13) Connor, J. N. L.; Shan, X.; Totenhofer, A. J.; Xiahou, C. New Theoretical Methods for Understanding Chemical Reactions in the Energy and Time Domains. In Multidimensional Quantum Mechanics with Trajectories; Shalashilin, D. V., de Miranda, M. P., Eds.; Collaborative Computational Project on Molecular Quantum Dynamics (CCP6); Daresbury Laboratory: Warrington, U.K., 2009; pp 38−47. (14) Xiahou, C.; Connor, J. N. L.; Zhang, D. H. Rainbows and Glories in the Angular Scattering of the State-to-State F + H2 Reaction at Etrans = 0.04088 eV. Phys. Chem. Chem. Phys. 2011, 13, 12981− 12997. (15) Shan, X.; Connor, J. N. L. Semiclassical Glory Analyses in the Time Domain for the H + D2(vi = 0, ji = 0) → HD(vf = 3, jf = 0) + D Reaction. J. Chem. Phys. 2012, 136 (044315), 1−18. (16) Xiahou, C.; Connor, J. N. L. The 6Hankel Asymptotic Approximation for the Uniform Description of Rainbows and Glories in the Angular Scattering of State-to-State Chemical Reactions: Derivation, Properties and Applications. Phys. Chem. Chem. Phys. 2014, 16, 10095−10111. (17) Nussenzveig, H. M. The Science of the Glory. Sci. Am. 2012, 306 (No. 1), 68−73. (18) Connor, J. N. L.; McCabe, P.; Sokolovski, D.; Schatz, G. C. Nearside−farside Analysis of Angular Scattering in Elastic, Inelastic and Reactive Molecular Collisions. Chem. Phys. Lett. 1993, 206, 119− 122. (19) McCabe, P.; Connor, J. N. L. Nearside−farside Analysis of Differential Cross Sections: Diffraction and Rainbow Scattering in Atom−Atom and Atom−Molecule Rotationally Inelastic Sudden Collisions. J. Chem. Phys. 1996, 104, 2297−2311. (20) Neumark, D. M.; Wodtke, A. M.; Robinson, G. N.; Hayden, C. C.; Lee, Y. T. Molecular Beam Studies of the F + H2 Reaction. J. Chem. Phys. 1985, 82, 3045−3066. (21) Wang, X.; Dong, W.; Qiu, M.; Ren, Z.; Che, L.; Dai, D.; Wang, X.; Yang, X.; Sun, Z.; Fu, B.; Lee, S.-Y.; Xu, X.; Zhang, D. H. HF(v′ = 3) Forward Scattering in the F + H2 Reaction: Shape Resonance and Slow-Down Mechanism. Proc. Nat. Acad. Sci. U.S.A. 2008, 105, 6227− 6231. (22) Hu, W.; Schatz, G. C. Theories of Reactive Scattering. J. Chem. Phys. 2006, 125 (132301), 1−15. (23) Karplus, M.; Tang, K. T. Quantum-mechanical Study of H + H2 Reactive Scattering. Discuss. Faraday Soc. 1967, 44, 56−67. (24) Miller, W. B.; Safron, S. A.; Herschbach, D. R. Exchange Reactions of Alkali Atoms with Alkali Halides: A Collision Complex Mechanism. Discuss. Faraday Soc. 1967, 44, 108−122. (25) Connor, J. N. L.; Child, M. S. Differential Cross Sections for Chemically Reactive Systems. Mol. Phys. 1970, 18, 653−679. (26) Connor, J. N. L. Resonance Regge Poles and the State-to-State F + H2 Reaction: QP Decomposition, Parameterized S Matrix, and Semiclassical Complex Angular Momentum Analysis of the Angular Scattering. J. Chem. Phys. 2013, 138 (124310), 1−21. (27) Dobbyn, A. J.; McCabe, P.; Connor, J. N. L.; Castillo, J. F. Nearside−farside Analysis of State Selected Differential Cross Sections for Reactive Molecular Collisions. Phys. Chem. Chem. Phys. 1999, 1, 1115−1124. (28) Anni, R.; Connor, J. N. L.; Noli, C. Improved Nearside−farside Method for Elastic Scattering Amplitudes. Phys. Rev. C: Nucl. Phys. 2002, 66 (044610), 1−11.

(29) Anni, R.; Connor, J. N. L.; Noli, C. Improved Nearside−farside Decomposition of Elastic Scattering Amplitudes. Khim. Fiz. 2004, 23 (No. 2), 6−12. Also available at: http://arXiv.org/abs/physics/ 0410266. (30) Connor, J. N. L.; Anni, R. Local Angular Momentum-local Impact Parameter Analysis: A New Tool for Understanding Structure in the Angular Distributions of Chemical Reactions. Phys. Chem. Chem. Phys. 2004, 6, 3364−3369. (31) Totenhofer, A. J.; Noli, C.; Connor, J. N. L. Dynamics of the I + HI → IH + I Reaction: Application of Nearside-farside, Local Angular Momentum and Resummation Theories using the Fuller and Hatchell Decompositions. Phys. Chem. Chem. Phys. 2010, 12, 8772−8791. (32) Sokolovski, D. Weak Values, “Negative Probability” and the Uncertainty Principle. Phys. Rev. A 2007, 76 (042125), 1−13. (33) Rowley, N. An l-window Formalism for Elastic Heavy-ion Scattering. J. Phys. G: Nucl. Phys. 1980, 6, 697−722. (34) Connor, J. N. L.; Thylwe, K.-E. Theory of Large Angle Elastic Differential Cross Sections for Complex Optical Potentials: Semiclassical Calculations using Partial Waves, l−windows, Saddles and Poles. J. Chem. Phys. 1987, 86, 188−195. (35) Schatz, G. C.; Amaee, B.; Connor, J. N. L. The Centrifugal Sudden Distorted Wave Method for Calculating Cross Sections for Chemical Reactions: Angular Distributions for Cl + HCl → ClH + Cl. Comput. Phys. Commun. 1987, 47, 45−53. (36) Vrinceanu, D.; Msezane, A. Z.; Bessis, D.; Connor, J. N. L.; Sokolovski, D. Padé Reconstruction of Regge Poles from Scattering Matrix Data for Chemical Reactions. Chem. Phys. Lett. 2000, 324, 311−319. (37) Monks, P. D. D.; Xiahou, C.; Connor, J. N. L. Local Angular Momentum-Local Impact Parameter Analysis: Derivation and Properties of the Fundamental Identity, with Applications to the F + H2, H + D2 and Cl + HCl Chemical Reactions. J. Chem. Phys. 2006, 125 (133504), 1−13. (38) Sokolovski, D.; De Fazio, D.; Cavalli, S.; Aquilanti, V. Overlapping Resonances and Regge Oscillations in the State-to-state Integral Cross Sections of the F + H2 Reaction. J. Chem. Phys. 2007, 126 (121101), 1−5. (39) Fuller, R. C. Qualitative Behavior of Heavy-ion Elastic Scattering Angular Distributions. Phys. Rev. C: Nucl. Phys. 1975, 12, 1561−1574. (40) Xiahou, C.; Connor, J. N. L. A New Rainbow: Angular Scattering of the F + H2(vi = 0, ji = 0) → FH(vf = 3, jf = 3) + H Reaction. J. Phys. Chem. A 2009, 113, 15298−15306. (41) Stark, K.; Werner, H.-J. An Accurate Multireference Configuration Interaction Calculation of the Potential Energy Surface for the F + H2 → HF + H Reaction. J. Chem. Phys. 1996, 104, 6515− 6530.

6573


Application of Heisenberg's S Matrix Program to the Angular

Recommend Documents