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Theory of forward glory scattering for chemical reactions

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

The theory of forward glory scattering is developed for a state-to-state chemical reaction whose scattering amplitude can be expanded in a Legendre partial wave series.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

Two transitional approximations are derived that are valid for angles on, and close to, the axial caustic associated with the glory.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

These are the integral transitional approximation (ITA) and the semiclassical transitional approximation (STA), which is obtained when the stationary phase method is applied to the ITA.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

Both the ITA and STA predict that the scattering amplitude for glory scattering is proportional to a Legendre function of real degree or, to a very good approximation, a Bessel function of order zero.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

A primitive semiclassical approximation (PSA) is also derived that is valid at larger angles, away from the caustic direction, but which is singular on the caustic.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

The PSA demonstrates that glory structure arises from nearside–farside (NF) interference, in an analogous way to the two-slit experiment.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

The main result of the paper is a uniform semiclassical approximation (USA) that correctly interpolates between small angles, where the ITA and STA are valid, and larger angles where the PSA is valid.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

The USA expresses the scattering amplitude in terms of Bessel functions of order zero and unity, together with N and F cross sections and phases.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

In addition, various subsidiary approximations are derived.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

The input to the theory consists of accurate quantum scattering matrix elements.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

The theory also has the important attribute that it provides physical insight by bringing out semiclassical and NF aspects of the scattering.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met1

The theory is used to show that the enhanced small angle scattering in the F + H₂(v_i = 0, j_i = 0, m_i = 0) → FH(v_f = 3, j_f = 3, m_f = 0) + H reaction is a forward glory, where v_i, j_i, m_i and v_f, j_f, m_f are initial and final vibrational, rotational and helicity quantum numbers respectively.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

The forward angle scattering for the H + D₂(v_i = 0, j_i = 0, m_i = 0) → HD(v_f = 3, j_f = 0, m_f = 0) + D reaction is also analysed and shown to be a forward glory, in agreement with a simpler treatment by D. Sokolovski (Chem. Phys. Lett., 2003, 370, 805), which is a special case of the STA.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

Introduction

The differential cross section for a state-to-state reactive molecular collision often exhibits a complicated interference pattern.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

An important problem, which is discussed in recent reviews,^1–13 is to analyse this scattering pattern in order to obtain information on the reaction dynamics.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

One theoretical approach for understanding structured angular distributions, and which is used in this paper, applies a nearside–farside (NF) decomposition to the partial wave series (PWS) representation of the scattering amplitude.^{2,5,12,14–31}

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

The aim of a NF analysis is to decompose the PWS exactly into the sum of two subamplitudes that have simpler properties, namely a N PWS subamplitude and a F PWS subamplitude.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

The complicated interference structure in the angular distribution can then often be interpreted as arising from interference between the N and F subamplitudes, or from the N subamplitude, or from the F subamplitude.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met1

NF theory has been developed for PWS expanded in a basis set of Legendre polynomials, associated Legendre functions or reduced rotation matrix elements.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

Applications have reported for the reactions: Cl + HCl → ClH + Cl,^14,16,29,30 F + H₂ → FH + H,²² H + D₂ → HD + H,²² F + HD → FH + D (or FD + H),²² and I + HI → IH + I.^25,26,31

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

The F + H₂ reaction provides a good example of the physical insight supplied by a NF analysis: the NF decomposition of the scattering amplitude for a state-to-state angular distribution suggested²² that the oscillatory forward angle scattering arises from a short-lived decaying Regge state with a lifeangle of ≈30°–40°.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met1

This suggestion that a short-lived resonance state contributes to the forward angle scattering was contrary to conventional wisdom at the time it was made (see, e.g., ^{ref. 32}) but was later verified by Sokolovski et al.^33,34

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

A second, and more fundamental, theoretical procedure for understanding structure in an angular distribution is to take the ħ → 0 limit of the scattering amplitude.³⁵

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met2

The scattering pattern is again interpreted as the interference between a small number of simpler (semiclassical) subamplitudes.^{2,29–31,33–38}

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met2

Unfortunately, carrying out a rigorous (uniform) asymptotic analysis of the scattering amplitude for a reactive collision is very difficult: it remains an important unsolved problem at the present time.

Type: Method | Advantage: No | Novelty: Old | ConceptID: Met2

However, progress has been made for a Legendre PWS whose scattering matrix element, when analytically continued, possesses poles in the complex angular momentum (CAM) plane.^{2,29–31,33–38}

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

A common observation in the angular distributions of state-to-state reactions is an enhancement of the forward angle scattering, which is often accompanied by oscillations.^1–13

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

The semiclassical theory developed in ^{refs. 29 and 30} for zero helicity state-to-state transitions shows that the forward angle scattering is associated with the factor 1/sin θ_R, where θ_R is the reactive scattering angle.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

The factor 1/sin θ_R is characteristic of an axial caustic,^39–43 which suggests that the enhanced intensity at forward angles is an example of glory scattering.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

Unfortunately, the semiclassical CAM theory presented in ^{refs. 29 and 30} is not uniformly valid at θ_R = 0.

Type: Method | Advantage: No | Novelty: Old | ConceptID: Met3

Sokolovski has partially overcome this defect in a letter whose main emphasis is the relation between CAM pole structure and forward angle scattering.³⁸

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

In particular, Sokolovski has developed a transitional semiclassical approximation, based on one given by Landau and Lifshitz (^{ref. 44}, p. 523), which he has applied to the H + D₂ reaction,³⁸ obtaining good agreement with the exact angular distribution at small values of θ_R.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met4

The present paper extends this research of Sokolovski,³⁸ and makes the following contributions to understanding glories in the forward angle scattering of reactive collisions:

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

(1) Two new transitional approximations are derived: the integral transitional approximation (ITA) and the semiclassical transitional approximation (STA), which is a stationary-phase approximation to the ITA.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

A further approximation to the STA yields the glory formula used by Sokolovski.³⁸

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

Transitional approximations are valid for angles on, and close to, the axial caustic, i.e., for θ_R ≈ 0, but they become increasingly inaccurate at larger angles.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

(2) A primitive semiclassical approximation (PSA) is obtained, which is valid away from the caustic direction at larger angles, but which is singular (infinite) at θ_R = 0.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

The PSA angular distribution is analogous to the interference pattern in the famous two-slit experiment and clearly demonstrates that the forward-angle glory scattering arises from NF interference.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

If the interference term in the PSA is neglected, we obtain the classical semiclassical approximation (CSA), which lets us make contact with quasiclassical trajectory calculations of the angular scattering.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

(3) A uniform semiclassical approximation (USA) is derived which correctly interpolates between θ_R ≈ 0, where the transitional approximations are valid, and larger angles where the PSA is valid.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

The derivation of the USA makes use of mathematical techniques for the (uniform) asymptotic evaluation of oscillating integrals.^45–51

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met5

The ITA, STA, PSA, CSA and USA, together with some subsidiary approximations, are derived in section II.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

An important property of all these approximations is that the semiclassical picture is clearly evident, even though their input consists of accurate scattering matrix elements, i.e. it is not necessary to assume the Wentzel–Kramers–Brillouin (WKB) approximation for the scattering matrix elements (although this approximation, and other ones, could be used if convenient).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

The theory of section II is applied in sections III and IV to two benchmark state-to-state reactions that exhibit forward angle scattering.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa2

These are the F + H₂ → FH + H and H + D₂ → HD + D reactions respectively.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa2

Section V contains our conclusions.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

The appendix examines five different ways to approximate a Legendre function by a Bessel function of order zero that have been used in the molecular collision theory literature.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa3

Theory of forward glory scattering for reactive molecular collisions

Introduction

We consider a state-to-state chemical reaction of the type A + BC(v_i, j_i, m_i = 0) → AB(v_f,j_f,m_f = 0) + Cwhere v_i, j_i, m_i and v_f, j_f, m_f are the vibrational, rotational and helicity quantum numbers for the initial and final states, respectively.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

Because the helicity quantum numbers are equal to zero, the scattering amplitude, f(θ_R), can be written as a Legendre partial wave series (PWS) where k is the initial translational wavenumber, J is the total angular momentum quantum number, S_J is a scattering matrix element, P_J(•) is a Legendre polynomial of degree J and θ_R ∈ [0,π] is the reactive scattering angle, i.e., the angle between the outgoing AB molecule and the incoming A atom.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The subscript, v_i, j_i, m_i → v_f, j_f, m_f, has been omitted from f(θ_R) and S_J for notational simplicity, as has the subscript, v_i, j_i, from k.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

The series (1) also occurs in some approximate theories of chemical reactions that expand the PWS in a basis set of Legendre polynomials (see, for example, ^{refs. 14, 16, 25, 26, 29, 30, 52 and 53}).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

The differential cross section is given by σ(θ_R) = |f(θ_R)|².For the theoretical development that follows, it is convenient to first write the PWS (1) in the standard form with θ_R ∈ [0,π], where the argument of the Legendre polynomial is cos θ_R rather than cos(π – θ_R).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

The modified scattering matrix element S̃_J in eqn. (3) is defined by S̃_J = exp(iπJ)S_J, J = 0,1,2,…

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

We begin the derivation by applying the Poisson summation formula (e.g., ^{ref. 39}) to the PWS (3), which yields the exact result for θ_R ∈ [0,π) where λ = J + 1/2 and the notation S̃_J = S̃_λ–1/2 ≡ S̃(λ)has been used.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met6

In eqn. (4), S̃(λ) and P_λ–1/2(cos θ_R) have been interpolated (and extrapolated where necessary) from half-integer values of λ = 1/2,3/2,5/2,… to continuous values, λ ∈ [0,∞).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

Sections III and IV describe the interpolation used for S̃(λ).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

In practice, separate interpolations are used for |S̃(λ)| and arg S̃(λ), where arg does not in general denote the principal value, rather arg S̃(λ) is a continuous function with arg S̃(λ) ∈ (–∞,∞).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

In eqn. (4), P_λ–1/2(cos θ_R) is a Legendre function of the first kind of real degree, λ – 1/2, with θ_R ∈ [0,π).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

[n.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

b., P_λ–1/2(cos θ_R) is singular at θ_R = π, except for λ = 1/2,3/2,5/2,…]

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

Under semiclassical conditions, kR ≫ 1, where R is the reaction radius, the integrands in the series (4) are highly oscillatory functions.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

We assume, as is commonly the case,^39–41 that any points of stationary phase occur only for the m = 0 integrand.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

We then have for θ_R ∈ [0,π)The upper limit, λ_max + 1/2 = J_max + 1, on the integral in eqn. (6) recognises the fact that the PWS (3) has, in practice, a finite number of non-zero terms,⁵⁴ with S̃_J ≡ 0 for J > J_max.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

Note: if we numerically evaluate the integral (6) and compare it with the result from summing the PWS (3), we can then test the accuracy of neglecting terms with m ≠ 0 in the Poisson series (4).

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp1

Eqn. (6) is the starting point for the approximations derived below.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

In the derivations, we shall often expand arg S̃(λ) about a real point λ = λ_s in a Taylor's series to second order arg S̃(λ) = arg S̃(λ_s) + (λ_s) (λ – λ_s) + ½′(λ_s) (λ – λ_s)²wherewill be called the quantum deflection function.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

The phrase “deflection function” indicates that (λ) is analogous to the classical deflection function often used in the semiclassical theory of elastic scattering.^39–41

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

The adjective “quantum” indicates that {S̃_J}, and its continuation S̃(λ), are in principle exact quantum quantities (although, if convenient, approximate scattering matrix elements could be used).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

In addition, the notations andhave been employed in the Taylor expansion (7).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

Figs. 1(a) and 1(b) show, for the F + H₂ reaction of section III, the behaviour of arg S̃(λ) and (λ) around the point λ = λ_g where arg S̃(λ) is stationary (a maximum) so that (λ_g) = 0.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

In addition, Fig. 1(b) illustrates the quantities λ_±(θ_R) and λ(θ_R;φ) with φ ∈ [0,π] which play an important rôle in the derivations below.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

Transitional approximations (TA)

In this section, two transitional approximations are derived for the forward glory scattering.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

A TA is valid on, and close to, the axial caustic, i.e., for values of θ_R close to θ_R = 0.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

Integral transitional approximation (ITA)

When θ_R = 0, we have P_λ–1/2(1) = 1 for all λ ∈ [0,∞).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

For θ_R close to θ_R = 0, we assume that P_λ–1/2(cos θ_R) is slowly varying relative to the oscillations of S̃(λ) and can be removed from the integrand of eqn. (6).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

The (real) value used for λ in the removal of P_λ–1/2(cos θ_R) is that given by the stationary phase condition (λ_g) = 0which defines the glory angular momentum variable, λ_g, see Fig. 1.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

In practice, there may be more than one root to eqn. (9)-see sections III and IV for examples.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp2

In this situation, we use the λ_g that makes the largest contribution to the integral in eqn. (6), when the stationary phase approximation is employed (see section IIB.2).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

We can now approximate eqn. (6) by f_ITA(θ_R) = GP_{λ_g–1/2}(cos θ_R)where G denotes the complex-valued integralThe corresponding differential cross section isσ_ITA(θ_R) = |G|²P_{λ_g–1/2}(cos θ_R)²Eqns. (10)–(12) define the ITA for forward glory scattering.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

Note that the value of the cross section at θ_R = 0 is finite, being σ_ITA(θ_R = 0) = |G|².

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

Semiclassical transitional approximation (STA)

The ITA can be further approximated if we evaluate the integral (11) by the stationary phase method.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

We again assume that the stationary phase condition is given by eqn. (9) and that the situation in Fig. 1 applies.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

Then with the help of the stationary phase result [^{ref. 55}, eqns. (4.1) and (4.12)] where F′(λ_s) = 0, a < λ_s < b, G(λ_s) ≠ 0,∞ and F″(λ_s) ≠ 0,∞, we obtain for the STA The corresponding differential cross section is.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

Eqns. (14) and (15) define the STA for forward glory scattering.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

At θ_R = 0, the value of the cross section is Note that eqn. (7) becomes for this case arg S̃(λ) = arg S̃(λ_g) + ½′(λ_g) (λ – λ_g)²where ′(λ_g) is negative, i.e., arg S̃(λ) has a maximum at λ = λ_g as illustrated in Fig. 1.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

Remarks

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

(i) The ITA is a global approximation, in that the input consists of {S̃_J}, and its continuation S̃(λ), which contain all the information on the dynamics of the reaction.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

In contrast, the STA is a local approximation, since it only requires dynamical information at λ = λ_g.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

(ii) Both the ITA and STA predict that the angular distribution for forward glory scattering is proportional to P_{λ_g–1/2}(cos θ_R)².

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

However conversely, if we find that the angular scattering at θ_R ≈ 0 can be fitted to the form constant × P_{λ_t–1/2}(cos θ_R)² say, then it does not follow that we have a forward glory without further investigation, e.g., we must check that (λ_t) = 0 is satisfied.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp3

For there may be other dynamical mechanisms that result in the integral (11) being dominated by a small range of λ values around λ = λ_t, but with the property, (λ_t) ≠ 0.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp3

(iii) Following Ford and Wheeler,^56,57 it has been common in the semiclassical theory of glory scattering for elastic collisions to approximate Legendre polynomials by Bessel functions of order zero.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod8

Eqns. (12) and (15) demonstrate that this approximation is unnecessary [this last result has also been obtained by Child, but not commented on: see eqn. (5.52) of ^{ref. 40}].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod8

The appendix examines five ways in which a Legendre function can be approximated by a Bessel function of order zero.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa3

The Hilb approximation is the one recommended for use: which has an error, O(λ^–3/2), that is uniform with respect to θ_R ∈ [0,π – ε] with ε > 0.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

At θ_R = 0, where eqn. (17) takes the indeterminate form 0/0, we use the well-known limit, θ_R/sinθ_R → 1 as θ_R → 0.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

When the Hilb relation (17) is used to replace the Legendre functions in the ITA and STA, the resulting approximations will be denoted ITA/J₀ and STA/J₀ respectively.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

If we make the additional approximation of replacing the factor (θ_R/sinθ_R)^1/2 by unity for θ_R ≈ 0 in the STA/J₀, we obtain the forward glory approximation used by Sokolovski-see eqn. (17) of .^{ref. 38}

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

100

(iv) The ITA and STA both predict that the cross section is zero at the roots of the equation, P_{λ_g–1/2}(cos θ_R) = 0.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

101

Similarly the ITA/J₀ and STA/J₀ predict zero cross sections at the roots of J₀(λ_gθ_R) = 0 [note that the first five roots of J₀(x) = 0 are x = 2.405, 5.520, 8.654, 11.792 and 14.931, from ^{ref. 58}. p. x, table II).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

Primitive semiclassical approximation (PSA)

102

As θ_R moves away from θ_R = 0, the assumption of the ITA and STA that P_λ–1/2(cos θ_R) is slowly varying becomes less valid.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

103

Instead, we must use the asymptotic approximation which is valid for λ sinθ_R ≫ 1.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

104

The two exponentials in eqn. (18) have a physical interpretation: The term exp(+iλθ_R) describes an angular wave travelling anti-clockwise in θ_R and corresponds to farside (F) scattering in a semiclassical treatment, whilst the term exp(–iλθ_R) represents an angular wave travelling clockwise in θ_R and corresponds to nearside (N) scattering.^14–30

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

105

Substituting approximation (18) into the integral (5) lets us write f(θ_R) = f₊(θ_R) + f_–(θ_R) where the F and N subamplitudes are with

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

106

Next we apply the stationary phase approximation to the I_±(θ_R) integrals, assuming that |S̃(λ)| is slowly varying.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

107

The stationary phase condition is or (λ) = ∓θ_R for λ = λ_±(θ_R) where we have assumed that there exist a unique pair of real valued roots to eqn. (23).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

108

The λ_±(θ_R) are illustrated in Fig. 1(b).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

109

Note that λ_–(θ_R) and λ₊(θ_R) are coincident at θ_R = 0, since λ_g = λ_±(θ_R = 0).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

110

Also Fig. 1(b) shows that ′(λ_±(θ_R)) < 0.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

111

Explicit formulae for the λ_±(θ_R) can be obtained when the quadratic approximation (16) is made for arg S̃(λ).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

112

For (λ), we have (λ) = –|′(λ_g)|(λ – λ_g) and the stationary phase eqn. (23) then gives

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

113

Applying the stationary phase approximation (13) to the integrals (21) yields The term exp[+iλ_±(θ_R)θ_R] in eqn. (25) describes an angular wave travelling anti-clockwise in θ_R, so that f₊(θ_R) corresponds to the F scattering, whilst the term exp[–iλ_±(θ_R)θ_R] represents an angular wave travelling clockwise in θ_R, so that f_–(θ_R) corresponds to the N scattering.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

114

The formulae for the f_±(θ_R) subamplitudes take a simpler form if we first define the F and N cross sections The σ_±(θ_R) are analogous to classical cross sections.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

115

This analogy can be made more explicit by defining the impact parameter variable, b = λ/k together with b_±(θ_R) = λ_±(θ_R)/k.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

116

We can then write eqn. (26) in the form Substituting eqns. (25) and (26) into eqn. (20), we obtain for the F and N subamplitudes f₊(θ_R) = –[σ₊(θ_R)]^1/2exp[iβ₊(θ_R)]andf_–(θ_R) = –i[σ_–(θ_R)]^1/2exp[iβ_–(θ_R)]where the F and N phases β_±(θ_R) are defined byβ_±(θ_R) = arg S̃(λ_±(θ_R)) ± λ_±(θ_R)θ_R.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

117

The full PSA scattering amplitude is then given by f_PSA(θ_R) = –[σ₊(θ_R)]^1/2 exp[iβ₊(θ_R)] – i[σ_–(θ_R)]^1/2 exp[iβ_–(θ_R)].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

118

The extra factor i in eqn. (28) compared to eqn. (27), arises from the phase difference exp(iπ/2) between the F and N angular waves in eqn. (18).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

119

Eqns. (26)–(30) define the PSA for the scattering amplitude.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

120

The corresponding PSA differential cross section for glory scattering is given by σ_PSA(θ_R) = σ₊(θ_R) + σ_–(θ_R) + 2[σ₊(θ_R)σ_–(θ_R)]^1/2sin[β₊(θ_R) – β_–(θ_R)].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

121

Eqn. (31) has the typical form of a two slit interference pattern for the F and N scattering.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

Classical semiclassical approximation (CSA)

122

If the sinusoidal term in eqn. (31) is neglected, we obtain a result that will be called the CSA σ_CSA(θ_R) = σ₊(θ_R) + σ_–(θ_R)σ_CSA(θ_R) can be estimated by quasiclassical trajectory (QCT) calculations.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod13

123

If the |S̃(λ_±(θ_R))|² in eqn. (26) contain a sizable contribution from tunnelling, then we might expect the QCT method to underestimate σ_CSA(θ_R).

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp4

124

On the other hand, the QCT binning procedure used to “quantize” the final states, has a weak theoretical justification, and could result in an overestimate of σ_CSA(θ_R).

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp5

125

It is interesting to take the θ_R → 0 limit of eqn. (32).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod14

126

We find The singularity at θ_R = 0 can be clearly seen in eqn. (33).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod14

127

Its origin is the (non-uniform) asymptotic approximation (18), which is not valid for θ_R → 0.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod14

128

Eqn. (33) is analogous to the classical result for elastic scattering from a central potential.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res13

129

[n.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

130

b., there are errors in the formulae given by Pauly in eqn. (2.15) of ^{ref. 59} and in eqn. (42) of ^{ref. 60}].

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

131

We can also derive eqn. (33) from the STA intensity (15), by averaging over the oscillations in P_{λ_g–1/2}(cos θ_R)².

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

132

With the help of eqn. (18), we find 〈P_{λ_g–1/2}(cos θ_R)²〉 ≈ 1/(πλ_gsin θ_R) so that eqn. (15) reduces to eqn. (33).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

133

An alternative derivation of eqn. (33) uses the Hilb approximation (17) together with the average 〈J₀(λ_gθ_R)²〉 ≈ 1/(πλ_gθ_R).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod16

Uniform semiclassical approximation (USA)

Introduction

134

The ITA and STA do not merge smoothly with the PSA when θ_R increases from θ_R = 0 to larger angles.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

135

We can rectify this defect by deriving a USA which does change smoothly as θ_R moves away from the axial caustic and which contains the ITA/J₀ and PSA as limiting cases.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod17

136

In particular, we generalise to reactive collisions the uniform semiclassical treatment of forward glory elastic scattering given by Berry⁴⁵ (see also ^{refs. 46–51} for more general uniform approximations).

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

137

The derivation of the USA proceeds by applying uniform asymptotic techniques to a two-dimensional integral representation for f(θ_R).

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met5

138

The first step in the derivation inserts the Hilb approximation (17) for P_λ–1/2(cos θ_R) into the integrand of eqn. (5) giving Now the Bessel function, J₀(x), has the following integral representation, [^{ref. 58}, p. 57, eqn. (4.3), together with J₀(x) = J₀(–x)].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod17

139

In the second step, we insert eqn. (35) into eqn. (34) obtaining the following two-dimensional integral for f(θ_R) where the phase is defined by B(θ_R;λ,φ) = arg S̃(λ) – λθ_Rcosφ.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod17

140

Before proceeding, it is first helpful to examine the stationary points of B(θ_R;λ,φ).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

Stationary points of B(θ_R;λ,φ)

141

The stationary phase equations for the two dimensional integral (36) are where eqn. (8) has been used.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

142

For θ_R ≠ 0 the two roots of eqn. (39) for φ ∈ [0,π] (i.e., for φ in the range of integration) are φ = 0 and φ = π.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

143

Substituting the root φ = 0 into eqn. (38) gives (λ_–(θ_R)) = +θ_R whilst substituting the root φ = π into eqn. (38) leads to (λ₊(θ_R)) = –θ_R, which agree with eqn. (23).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

144

The stationary points of B(θ_R;λ,φ) are therefore (λ_–(θ_R),φ = 0) and (λ₊(θ_R),φ = π).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

145

The values of B(θ_R;λ,φ) at the stationary points are B(θ_R;λ_±(θ_R),φ = π,0) = arg S̃(λ_±(θ_R)) ± λ_±(θ_R)θ_R ≡ β_±(θ_R)where the definition of the β_±(θ_R) in eqn. (29) has been used.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

146

It is convenient to continue to assume θ_R ≠ 0 in the following calculations of stationary phase points.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

Stationary phase integration over λ

147

The phase B(θ_R;λ,φ), defined by eqn. (37), is a rapidly varying function of λ.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

148

It is therefore legitimate to apply the stationary phase approximation to the integral over λ in eqn. (36).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod19

149

There is one stationary point, λ = λ(θ_R;φ), given by Note that λ(θ_R;φ) varies from λ(θ_R;φ = 0) ≡ λ_–(θ_R) to λ(θ_R;φ = π) ≡ λ₊(θ_R) as φ changes from 0 to π, as illustrated in Fig. 1(b).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod19

150

Applying the stationary phase formula (13) to eqn. (36), and noting that together with ′(λ(θ_R;φ)) < 0 [see Fig. 1(b)], we obtain Remark:

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod19

151

When θ_R ≈ 0, the phase B(θ_R;λ(θ_R;φ),φ) in eqn. (42) is a slowing varying function of φ, so that it is not legitimate to apply the stationary phase approximation again.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp6

152

However, as θ_R moves to larger angles, B(θ_R;λ(θ_R;φ),φ) varies more rapidly and we can once more apply the stationary phase technique, which provides us with a check on the PSA result (30).

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp6

153

The stationary phase condition is which simplifies, with the help of eqn. (41), to The two stationary points of eqn. (44) are φ = 0 and φ = π for φ ∈ [0,π].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod20

154

This means that the stationary points coincide with the end points of integration in eqn. (42), so the stationary phase result, eqn. (13), is not immediately applicable, since it assumed a < λ_s < b.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

155

However, the modification when a stationary point coincides with an end point is simple: the result in eqn. (13) is multiplied by 1/2.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

156

[^{ref. 55}, p. 77].

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

157

We then find that the stationary phase evaluation of the integral (42) results in the PSA, eqn. (30) for f(θ_R).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res17

Exact change of variable

158

When θ_R ≈ 0, the phase of the integrand in eqn. (42), namely B(θ_R;λ(θ_R;φ),φ) = arg S̃(λ(θ_R;φ)) – λ(θ_R;φ)θ_Rcosφwould be the same as the phase of the Bessel function of order zero, eqn. (35), were it not for the slow dependence of λ(θ_R;φ) on φ.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp7

159

This suggests that we make the following exact one-to-one change of variableB(θ_R;λ(θ_R;φ),φ) = A(θ_R) – ζ(θ_R)cosψ where the new parameters A(θ_R) and ζ(θ_R) are functions of the old parameter θ_R, whilst the new variable ψ = ψ(θ_R;φ) is a function of both θ_R and the old variable φ [and inversely, φ = φ(θ_R;ψ)].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

160

The rhs of eqn. (45) can also be regarded as the unfolding of a germ of infinite codimension.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

161

The stationary phase condition for the lhs of eqn. (45) has already been written down in eqns. (43) and (44).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

162

The two stationary points are φ = 0 and φ = π for φ ∈ [0,π].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

163

For the rhs of eqn. (45), the stationary phase condition is ζ(θ_R)sinψ = 0 and the two stationary points are ψ = 0 and ψ = π for ψ ∈ [0,π].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

164

The requirement that the transformation (45) be one-to-one, and hence that dφ/dψ ≠ 0,∞, is used to determine A(θ_R) and ζ(θ_R).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

165

Differentiating both sides of eqn. (45) with respect to ψ and using the lhs of eqn. (44) gives In order that dφ/dψ ≠ 0,∞ in eqn. (46), we evidently must have the correspondences.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

166

Note that dφ/dψ in eqn. (46) becomes 0/0, i.e., indeterminate, when eqns. (47) are substituted into it and a more detailed analysis is necessary to determine the explicit form of dφ/dψ at ψ = 0 and ψ = π.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

167

This is done in section IID.6 below.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

168

Inserting the correspondence φ = 0 ↔ ψ = 0 into eqn. (45) gives β_–(θ_R) = A(θ_R) – ζ(θ_R) where eqn. (40) has been used.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod22

169

Similarly, inserting the correspondence φ = π ↔ ψ = π into eqn. (45) yields β₊(θ_R) = A(θ_R) + ζ(θ_R).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod22

170

Solving the linear eqns. (48) and (49) results in A(θ_R) = ½[β₊(θ_R) + β_–(θ_R)] and ζ(θ_R) = ½[β₊(θ_R) – β_–(θ_R)].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod22

171

Eqns. (50) and (51) are the sought-for relations which express A(θ_R) and ζ(θ_R) in terms of the F and N phases, β_±(θ_R).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res19

172

Next we use eqn. (45) and the correspondences (47) to change variable from φ to ψ in the integral (42) for f(θ_R).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod23

173

We obtain where φ is to be regarded as a function of ψ, i.e., φ = φ(θ_R;ψ).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod23

Approximation of the pre-exponential factor

174

Next we express the pre-exponential factor in eqn. (52) in the form{•} = p(θ_R) + q(θ_R)cosψ + r(θ_R;ψ)sinψinvolving the functions p(θ_R), q(θ_R) and r(θ_R;ψ).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod24

175

When θ_R ≈ 0, the factor {•} is approximately constant since λ(θ_R ≈ 0;φ) ≈ λ_g [see Fig. 1(b)], whereas when θ_R moves away from zero, the integrand becomes oscillatory, receiving its contributions from the two stationary points, ψ = 0 and ψ = π, where the third term on the rhs of eqn. (53) is zero.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod24

176

These considerations suggest that we neglect the term r(θ_R;ψ)sinψ in eqn. (53) and approximate the pre-exponential factor by with φ = φ(θ_R;ψ).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod24

177

The integral representation (35) for J₀(x) and the identity [^{ref. 58}, p. 2, eqn. (1.5)] lets us write eqns. (52) and (54) in terms of Bessel functions of order 0 and 1, namely

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod24

Determination of p(θ_R) and q(θ_R)

178

To determine p(θ_R) and q(θ_R), we substitute the correspondences φ = 0 ↔ ψ = 0 and then φ = π ↔ ψ = π into eqn. (54), obtaining andwhere the relations λ(θ_R;φ = 0) ≡ λ_–(θ_R) and λ(θ_R;φ = π) ≡ λ₊(θ_R) have also been used.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod25

179

The two linear eqns. (56) and (57) have solutions andFinally we must obtain explicit expressions for dφ/dψ at ψ = 0 and ψ = π.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod25

180

To do this we rewrite eqn. (46) in the form and differentiate with respect to ψ, obtaining.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod25

181

Substituting the correspondences φ = 0 ↔ ψ = 0 and φ = π ↔ ψ = π into eqn. (60), then gives and where the results λ(θ_R;φ = 0) ≡ λ_–(θ_R) and λ(θ_R;φ = π) ≡ λ₊(θ_R) have again been used.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod25

182

Substituting eqns. (61) and (62) into eqns. (58) and (59) and using eqns. (26), lets us write p(θ_R) and q(θ_R) in the form and

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod25

Uniform approximation

183

The USA scattering amplitude, f_USA(θ_R), is obtained upon combining eqns. (55), (63), (64) where A(θ_R) and ζ(θ_R) are defined by eqns. (50) and (51) respectively.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod26

184

The corresponding USA differential cross section is given by.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod26

185

Eqns. (65) and (66) express the USA scattering amplitude and cross section in terms of the F and N cross sections, σ_±(θ_R), and phases, β_±(θ_R).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod26

186

Now the equations J₀(x) = 0 and J₁(x) = 0 have an infinite number of simple roots (zeros), with no roots in common (^{ref. 58}, p. 105).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res20

187

This implies that σ_USA(θ_R), in contrast to the ITA and STA, is never equal to zero unless at some θ_R, we have σ₊(θ_R) = σ_–(θ_R) and J₀(ζ(θ_R)) = 0.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res20

Limiting cases

188

We must also check that eqn. (65) contains as limiting cases the non-uniform PSA and STA/J₀ scattering amplitudes.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp8

189

The PSA is valid when θ_R moves to larger angles, where we have ζ(θ_R) ≫ 1.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp8

190

In this limit, we can replace the two Bessel functions in eqn. (65) by their asymptotic forms [^{ref. 58}, p. 83, eqn. (5.34)] and [^{ref. 58}, p. 83, eqn. (5.36), together with J₁(x) = –dJ₀(x)/dx] We then find, with the help of eqns. (50) and (51), that the USA scattering amplitude, eqn. (65), reduces to the PSA scattering amplitude, eqn. (30).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res21

191

Next we consider the STA/J₀, which is valid for θ_R → 0.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp8

192

We make the following three approximations to deduce the limiting form of eqn. (65) when θ_R → 0:

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

193

(a) Since λ_±(θ_R) → λ_g [see, Fig. 1(b)] and therefore, σ₊(θ_R) → σ_–(θ_R) when θ_R → 0, we can neglect the term containing J₁(ζ(θ_R)) in eqn. (65).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

194

[N. b., we also have ζ(θ_R) → 0 for the argument of the Bessel function].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

195

(b) For A(θ_R), we similarly find that A(θ_R) → arg S̃(λ_g) as θ_R → 0.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

196

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

197

We can find it using the quadratic approximation (16) for arg S̃(λ) together with the definitions (29) and (51) for β_±(θ_R) and ζ(θ_R) respectively.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

198

The F and N angular momentum variables are given by eqn. (24), namely We then find that the F and N phases in the quadratic approximation are β_±(θ_R) = arg S̃(λ_g) + ½θ2R/|′(λ_g)| ± λ_gθ_Rwhich shows that ζ(θ_R) → λ_gθ_R as θ_R → 0.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

199

Substituting these three limits into eqn. (65), we obtain which is the scattering amplitude for the STA/J₀, see eqns. (14) and (17).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

Forward glory scattering in the F + H₂ → FH + H reaction

200

The F + H₂ reaction is an example of a benchmark exoergic reaction that has been extensively studied by theory and experiment.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

201

In particular, the forward angle scattering associated with the FH(v_f = 3) final vibrational state has been of much interest, as discussed in the reviews, ^{refs. 1, 10, 11, 61 and 62} (for other recent research on this reaction, see .^{refs. 63–79})

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

202

In this section we apply the glory theory of section II to the angular scattering of the state-to-state transition F + H₂(v_i = 0, j_i = 0, m_i = 0) → FH(v_f = 3, j_f = 3, m_f = 0) + H

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa2

Input scattering matrix data

203

The input to the glory analysis consists of a set of accurate quantum scattering matrix elements, {S̃_J} for J = 0(1)23 (and {S̃_J ≡ 0} for J > 23), computed for the Stark-Werner potential energy surface⁸⁰ at a total energy of E = 0.3872 eV, measured with respect to the classical minimum of the H₂ potential energy curve.^22,33

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

204

The translational energy is 0.119 eV, which corresponds to a translational wavenumber of k = 10.2 Å^–1.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod28

205

Fig. 2(a) shows the modulus of the scattering matrix input data, |S̃_λ–1/2|, for λ = ½,,,…,23½ (solid circles), plotted versusλ, together with its real valued continuation, |S̃(λ)|, drawn as a solid line.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res22

206

The continuation is achieved by interpolating the discrete data (and extrapolating to λ = 0) with polynomials of degree 6.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod29

207

It can be seen that |S̃(λ)| possesses two peaks at λ ≈ 10 and λ ≈ 16, and a minimum at λ ≈ 135.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

208

Fig. 2(b) displays the analogous plot for the input phase data, arg S̃_λ–1/2versusλ (solid circles) for λ = ½,,,…,23½, as well as its continuation, arg S̃(λ) versusλ (solid curve).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res23

209

We recall from section IIA that arg does not denote the principal value, which would produce phases/rad ∈ (–π,π]; in fact it can be seen in Fig. 2(b) that arg S̃(λ)/rad ∈ (0,40).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res23

210

The phase arg S̃_λ–1/2 is calculated by adding multiples of 2π to the phases computed as principal values, until two successive values, arg S̃_λ–1/2 and arg S̃_λ+1/2 differ by less than π.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

211

The continuation again uses polynomials of degree 6.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod29

212

It can be seen that arg S̃(λ) has a maximum at λ ≈ 16.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

213

The region around the maximum in Fig. 2(b), from λ = 13½ to λ = 17½, has already been shown in Fig. 1(a) as an aid to the derivations in section II.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

Nearside–farside analysis

214

Before applying the glory theory of section II, it is first helpful to perform a NF analysis of the angular scattering.^14,15

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod30

215

The NF decomposition for the Legendre PWS (3) is (for θ_R ≠ 0,π) f(θ_R) = f_N(θ_R) + f_F(θ_R) where the NF subamplitudes are defined by in which Q_J(cos θ_R) is a Legendre function of the second kind.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod30

216

The corresponding NF angular distributions are given by σ_N,F(θ_R) = |f_N,F(θ_R)|².

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod30

217

Note: it is common practice in the literature to use a superscript “–” to represent the N scattering, and a superscript “+” for the F scattering; however this notation has not been employed in eqns. (67)–(69) to avoid confusion with the “±” used in the semiclassical NF theory of sect. IIB-IID.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

218

Fig. 3 shows the results of the NF analysis for the PWS scattering amplitude, eqn. (3).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res24

219

It displays log plots of σ(θ_R), σ_N(θ_R) and σ_F(θ_R) versusθ_R.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res24

220

At backward angles the scattering is N dominated, with the oscillations arising from NF interference.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res24

221

In this angular range, the main features of the scattering can be understood using simple classical-like models, such as the semiclassical optical model (which is an approximate N theory).^22,25

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod31

222

In contrast, the scattering at forward angles contains substantial contributions from both the N and F subamplitudes (with F dominant), resulting in pronounced NF oscillations.

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223

The change in mechanism for the forward and backward angle scattering occurs around θ_R = 40°, which implies that the glory theory of section II should only be used for θ_R ≲ 40° (and in particular, the accuracy of the USA glory equations is expected to degrade as θ_R increases from θ_R = 0° to θ_R ≈ 40°).

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224

The σ(θ_R), σ_N(θ_R) and σ_F(θ_R) cross sections plotted in Fig. 3 are similar in their structure (generic) to the corresponding angular distributions for the Cl + HCl → ClH + Cl reaction, which have been analysed in detail in ^{refs. 29 and 30} using complex angular momentum techniques.

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225

In particular, the mechanism for the forward angle scattering in the Cl + HCl reaction is the decay of a short-lived Regge state.^29,30

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac13

226

The genericity of the cross sections led Dobbyn et al²². to suggest that the same mechanism also occurs for the F + H₂ reaction.

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227

This suggestion was contrary to accepted wisdom at the time it was made,^61,81 but was subsequently verified by Sokolovski et al.^33,34

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228

Later the participation of short-lived resonance states to the forward angle scattering has been confirmed by Chao and Skodje⁷² using adiabatic approximations and spectral quantization, as well as by Aquilanti et al.⁶²

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229

In a time-dependent experiment (or calculation), the forward angle scattering will be time-delayed relative to the backward angle scattering, as discussed in more detail for the H + D₂ reaction in sect. IV.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac16

Glory analysis of forward scattering

230

The radius of the interaction region for the F + H₂ reaction is approximately R = 2 Å, so we have kR ≈ 20.

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231

This confirms that kR ≫ 1, which is the condition for the semiclassical theory of section II to be valid.

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232

We begin by examining the validity of retaining just the m = 0 integral in the Poisson series, eqn. (4), since this approximation is fundamental to the analysis of sect. II.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod32

233

For θ_R = 0, at which angle P_λ–1/2(cos 0) = 1 for all λ ∈ [0,∞), the PWS, eqn. (3) has the value f(0) = 0.213 + 0.293i Å, which agrees closely with the result, 0.215 + 0.291i Å, obtained by numerical quadrature of the integral in eqn. (6).

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234

More generally, the angular distribution computed from eqn. (6) by quadrature for θ_R ∈ [0°,90°] agrees with the PWS cross section, eqn. (2), to within graphical accuracy (for θ_R > 90°, integrals with m = ±1 in the Poisson series, eqn. (4), start to become important numerically).

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235

These tests therefore justify the neglect of integrals with m ≠ 0 in the Poisson series, eqn. (4), at forward angles.

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236

The quantum deflection function, (λ), defined by eqn. (8), plays a key role in the glory theory of section II.

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237

Fig. 4 shows a plot of (λ) versusλ for the F + H₂ reaction.

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238

It can be seen that eqn. (9) has a (glory) root at λ_g ≈ 16.0 together with additional roots at λ_g ≈ 18.4,20.3 and 23.2.

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239

However |S̃(λ)|² is very small at these additional roots, being ≈10^–4, ≈10^–6 and ≈10^–8 respectively; their contribution has therefore been neglected.

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240

The ITA, defined by eqns. (10)–(12), and the STA, defined by eqns. (14) and (15), require an accurate value for λ_g.

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241

This was obtained by fitting a polynomial of degree 4 to the input phase data, arg S̃_λ–1/2, for λ = 13½(1)17½, as illustrated in Fig. 1(a).

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242

We then find on differentiation and solving eqn. (9) that λ_g = 16.02.

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243

We also find that ′(λ_g) = –0.522.

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244

Fig. 5 compares the PWS cross section of eqn. (2) with the ITA angular distribution for the angular range, 0° ≤ θ_R ≤ 40°.

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245

It can be seen there is close agreement between the PWS and ITA angular distributions for the forward peak up to θ_R ≈ 10°; then the ITA becomes less accurate as θ_R increases.

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246

This loss of accuracy is expected since (a) the ITA is, by construction, a transitional approximation designed to be most reliable at θ_R = 0°, and (b) the change in mechanism discussed in section IIIB becomes increasingly important as θ_R moves towards θ_R = 40°.

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247

The ITA/J₀, STA and STA/J₀ approximations are not shown separately in Fig. 5 since they all agree closely with the ITA, as does the transitional approximation used by Sokolovski for the H + D₂ reaction i.e., STA/J₀ with replacement of the factor (θ_R/sinθ_R)^1/2 by unity (see also the discussion in the Appendix).

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248

Fig. 5 also displays the glory scattering predicted by the USA.

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249

It can be seen that the USA is more accurate than the ITA, in particular for the amplitudes of the subsidiary glory oscillations.

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250

The change in mechanism at θ_R ≈ 40° is likely to account, in part, for the discrepancies between the USA and PWS angular distributions as θ_R increases.

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251

The PSA agrees closely with the USA for θ_R ≳ 4°; it is not shown separately in Fig. 5.

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252

Finally, the CSA is also plotted in Fig. 5.

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253

As expected, it is monotonic and passes through the glory oscillations; the divergence of the CSA as θ_R → 0° can be clearly seen.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res33

Forward glory scattering in the H + D₂ → HD + D reaction

254

The H + D₂ collision system is another example of a benchmark chemical reaction that has received extensive attention from theory and experiment.

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255

In particular, a time-delayed mechanism for the forward angle scattering of the HD(v_f = 3, j_f = 0) final rovibrational state has been of much interest^{1,37,38,82,83} (for other recent research on this reaction, see .^{ref. 84–96})

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256

The next section applies the glory theory of section II to the angular scattering of the state-to-state transition H + D₂(v_i = 0, j_i = 0, m_i = 0) → HD(v_f = 3, j_f = 0, m_f = 0) + D

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa2

Input scattering matrix data

257

The input to the glory analysis consists of a set of accurate quantum scattering matrix elements, {S̃_J} for J = 0(1)30, with {S̃_J ≡ 0} for J > 30, computed for the Boothroyd–Keogh–Martin–Peterson potential energy surface⁹⁷ number 2 at a total energy of E = 2.00 eV, measured with respect to the classical minimum of the D₂ potential energy curve.

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258

The corresponding translational energy is 1.81 eV, with the translational wavenumber being k = 26.4 Å^–1.

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259

The scattering matrix elements were computed by a new time dependent plane wavepacket method^{82,83,98–100} and were kindly supplied to the author by Dr S. C. Althorpe.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met7

260

Fig. 6(a) shows the modulus input data, |S̃_λ–1/2|, for λ = ½,,,…,30½, plotted versusλ (solid circles), together with its real valued continuation, |S̃(λ)| (solid curve).

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261

Polynomials of degree three have been used to interpolate the discrete data (and to extrapolate to λ = 0).

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262

It can be seen that |S̃(λ)| possesses two pronounced maxima at λ ≈ 15.7 and λ ≈ 22.7, and two pronounced minima at λ ≈ 13.1 and λ ≈ 19.0.

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263

Thus the |S̃(λ)| plot for the H + D₂ reaction is more complicated than that for the F + H₂ system in Fig. 2(a).

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264

The plot for the input phase data, arg S̃_λ–1/2versusλ (solid circles) for λ = ½,,,…,30½, as well as its continuation, arg S̃(λ) versusλ (solid curve), is shown in Fig. 6(b).

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265

The continuation also uses polynomials of degree three.

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266

It can be seen that arg S̃(λ) has a maximum at λ ≈ 24.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs8

267

The noticeable drop in arg S̃_λ–1/2 on going from λ = 18½ to λ = 19½ is associated with the corresponding near zero in |S̃_λ–1/2| evident in Fig. 6(a).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res35

Nearside–farside analysis

268

Fig. 7 shows log plots of σ(θ_R), σ_N(θ_R) and σ_F(θ_R) versusθ_R.

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269

At backward angles the scattering is N dominated, with the weak oscillations arising from NF interference.

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270

Simple classical-like models, such as the semiclassical optical model—an approximate N theory—can be used to understand the main features of the scattering in this angular range.^22,25

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271

The semiclassical optical model assumes direct dynamics and works best for rebound reactions in which the backward angle scattering arises from repulsive interactions between the reactants.

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272

Fig. 7 shows that the scattering at forward angles contains significant contributions from both the N and F subamplitudes (with N dominant), resulting in strong NF oscillations.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

273

At around θ_R = 40°, there is a change in mechanism for the forward and backward angle scattering, which implies, similar to the F + H₂ case, that the glory theory of sect. II will become less accurate as θ_R increases towards 40°.

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274

Aoiz et al³⁷. and Sokolovski³⁸ have shown that the oscillatory forward angle scattering arises from short-lived decaying Regge states with lifeangles of ≈10°–20°.

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275

At the International Complutense Seminar on Quantum Reactive Scattering, held at San Lorenzo de El Escorial, Spain, 20–23 June 2003, Althorpe showed a movie of the H + D₂ reaction.^82,83

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276

This movie, which I viewed, showed a wavepacket (containing initial translational energies in the range 1.3–2.2 eV, equally weighted) as it evolved for a duration of 100 fs from the initial state, H + D₂(v_i = 0, j_i = 0), to the final state HD(v_f = 3, j_f = 0) + D.

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277

The movie showed that the HD(v_f = 3, j_f = 0) molecule is formed by two distinct reaction mechanisms: the backward angle scattering arises from a direct recoil mechanism, whilst the forward angle scattering is formed by an indirect mechanism, being delayed by about 25 fs.^82,83

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278

Subsequently, I had private viewings of the movie, courtesy of Dr Althorpe.

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279

I have also examined clips and stills from the movie.

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280

The NF analysis in Fig. 7 is consistent with the two mechanism picture of the reaction dynamics depicted in the movie.

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281

Thus N scattering of the reactants, which dominates backward angles, is clearly visible in the movie and corresponds to the direct recoil mechanism.

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282

In contrast, the N and F scattering in the angular range, θ_R ≲ 40°, which arise from decaying Regge states (surface waves),^37,38 correspond to the indirect time-delayed mechanism.

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283

Indeed, the movie shows surface-like waves propagating around a circle of radius 1.9 Å (approximately the size of the transition state region), which then decay into the forward direction.

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284

If we compare the NF analysis for the H + D₂ reaction in Fig. 7 with that for the F + H₂ reaction in Fig. 3, we see that they are qualitatively similar.

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285

Thus we expect that the discussion given above for the H + D₂ reaction will also largely apply to the F + H₂ system (and vice versa).

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp9

286

Note that one difference is that H + D₂ is N dominated for θ_R ≲ 40°, whereas the F + H₂ reaction is F dominated in this angular range.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res40

Glory analysis of forward scattering

287

The H + D₂ reaction has a radius of interaction of approximately R = 2 Å.

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288

Thus kR ≈ 50, which means that the semiclassical theory of section II is applicable.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod32

289

We first examine the validity of retaining just the m = 0 integral in the Poisson series (4).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod32

290

For θ_R = 0, where P_λ–1/2 (cos 0) = 1 for all λ ∈ [0,∞), the PWS (3) has the value f(0) = –0.0640 – 0.0765i Å, in close agreement with the value –0.0647 – 0.0763i Å, obtained from the integral (6) by numerical quadrature.

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291

In fact, the angular distribution computed by quadrature from eqn. (6) agrees with the PWS cross section (2) to within graphical accuracy for θ_R ∈ [0°,90°].

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292

As for the F + H₂ reaction, integrals with m = ±1 in the Poisson series (4) only become important numerically when θ_R > 90°.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp10

293

Thus, at forward angles, we can safely neglect integrals with m ≠ 0 in the Poisson series (4).

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294

Fig. 8 shows a plot of (λ) versusλ for the H + D₂ reaction.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs10

295

The plot is more irregular than the one for the F + H₂ example of Fig. 4.

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296

This may be because (a) the H + D₂ reaction is more complicated dynamically than is the F + H₂ system: compare Fig. 2 with Fig. 6, and/or (b) there are different numerical errors in the scattering matrix computations for the two reactions.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp11

297

Fig. 8 shows that eqn. (9) has a (glory) root at λ_g ≈ 24 together with additional roots at λ_g ≈ 18.5,19.6 and 29.7.

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298

The first two roots are associated with the near-zero of S̃(λ) at λ ≈ 19.0.

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299

The absolute values of the slope, |d(λ)/dλ|, are also very large at these two roots and their contributions in the stationary phase approximation of eqn. (25) will be small—they have been neglected in the following.

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300

The root near λ = 29.7 has |S̃(λ_g)|² ≈ 10^–8; its contribution has also been neglected.

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301

Next we apply the glory approximations derived in Section II.

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302

We recall that the ITA requires a value for λ_g, the STA requires values for λ_g and the slope of (λ) at λ = λ_g, whilst the USA, PSA and CSA require values for λ_±(θ_R) and the slopes of (λ) at λ = λ_±(θ_R) for θ_R ≲ 40°.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod35

303

The irregularities in the (λ) curve of Fig. 8 near λ ≈ 24 make it difficult to extract accurate values for these quantities.

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304

To avoid this difficulty, we interpolated the input phase data, arg S̃_λ–1/2 for λ = 21½(1)25½ to determine λ_g, and then approximated (λ) by a straight line—in effect the input data has been smoothed.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod36

305

This procedure resulted in λ_g = 24.19 and ′(λ_g) = –0.233.

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306

These values are very close to those obtained by Sokolovski, namely λ_g = 24.2 and ′(λ_g) = –0.24, using a smoothed Padé reconstruction.

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307

The straight-line approximation for (λ) is drawn as a dashed line in Fig. 8 for (λ)/deg ∈ [–40,40].

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308

Values for the λ_±(θ_R) needed in the USA, PSA and CSA are then given by eqn. (24).

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309

Fig. 9 compares the PWS cross section of eqn. (2) with the ITA, USA and CSA glory angular distributions for the angular ranges, 0° ≤ θ_R ≤ 10° and 10° ≤ θ_R ≤ 40° (two angular ranges are used in order to show more clearly the errors of the different approximations).

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310

There is close agreement between the PWS and ITA angular distributions for the forward peak up to θ_R ≈ 12°, with the ITA then becoming less accurate as θ_R increases.

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311

As for the F + H₂ example, this loss of accuracy is expected since (a) the ITA is a transitional approximation which is most reliable at θ_R = 0°, and (b) the change in mechanism discussed in sect. IVB becomes increasingly significant as θ_R → 40°.

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312

The ITA/J₀, STA and STA/J₀ curves have not been plotted in Fig. 9 since they all agree closely with the ITA curve, as does the transitional approximation used by Sokolovski,³⁸i.e., STA/J₀ with the factor (θ_R/sinθ_R)^1/2 replaced by unity (see the Appendix for some additional discussion).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res47

313

Fig. 9 also displays the glory scattering predicted by the USA.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res48

314

It can be seen that the USA agrees closely with the PWS and ITA cross sections at smaller angles, but is more accurate than the ITA at larger angles, in particular, for the amplitudes of the subsidiary glory oscillations.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res48

315

The change in mechanism at θ_R ≈ 40° is likely to account, in part, for the discrepancies between the USA and PWS angular distributions as θ_R increases.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res48

316

The PSA is not shown separately in Fig. 9 since it agrees closely with the USA for θ_R ≳ 3°.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res49

317

The cross section for the CSA in Fig. 9 is seen to be monotonic.

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318

It passes through the glory oscillations and is divergent as θ_R → 0°.

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319

Finally, we note that Althorpe has introduced^82,83,99,100 the concepts of a time-dependent scattering amplitude and a time dependent cross section, and has demonstrated their value for understanding the dynamics of reactions in his wavepacket calculations.

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320

The NF and glory theory developed in this paper can also be used to understand structure in these time dependent quantities, when the time-dependent scattering amplitude can be expanded in a basis set of Legendre polynomials.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

Conclusions

321

The paper has developed a theory of forward glory scattering for a state-to-state chemical reaction whose scattering amplitude can be written as a Legendre PWS.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

322

The most important result derived in the paper is the USA, which correctly interpolates between small angles where the ITA and STA are valid, and larger angles where the PSA is valid.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

323

The USA expresses the scattering amplitude in terms of N and F cross sections and phases, together with Bessel functions of order zero and unity.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

324

The PSA clearly demonstrates that glory structure arises from nearside–farside (NF) interference, in an analogous way to the two-slit experiment.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

325

The theory has the important attribute that it provides physical insight by bringing out semiclassical and NF aspects of the scattering, even though no WKB phases are used.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

326

Rather, the input to the theory consists of accurate quantum scattering matrix elements.

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327

The theory has been used to show that the enhanced small angle scattering in the F + H₂(v_i = 0, j_i = 0, m_i = 0) → FH(v_f = 3, j_f = 3, m_f = 0) + H reaction is a forward glory.

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328

And the same is also true for the H + D₂(v_i = 0, j_i = 0) → HD(v_f = 3, j_f = 0) + D reaction, in agreement with a simpler treatment by Sokolovski³⁸ which is a special case of the present theory.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

Introduction

Theory of forward glory scattering for reactive molecular collisions

Introduction

Transitional approximations (TA)

Integral transitional approximation (ITA)

Semiclassical transitional approximation (STA)

Remarks

Primitive semiclassical approximation (PSA)

Classical semiclassical approximation (CSA)

Uniform semiclassical approximation (USA)

Introduction

Stationary points of B(θR;λ,φ)

Stationary phase integration over λ

Exact change of variable

Approximation of the pre-exponential factor

Determination of p(θR) and q(θR)

Uniform approximation

Limiting cases

Forward glory scattering in the F + H2 → FH + H reaction

Input scattering matrix data

Nearside–farside analysis

Glory analysis of forward scattering

Forward glory scattering in the H + D2 → HD + D reaction

Input scattering matrix data

Nearside–farside analysis

Glory analysis of forward scattering

Conclusions

Stationary points of B(θ_R;λ,φ)

Determination of p(θ_R) and q(θ_R)

Forward glory scattering in the F + H₂ → FH + H reaction

Forward glory scattering in the H + D₂ → HD + D reaction