1
Theory of forward glory scattering for chemical reactions

2
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.

3
Two transitional approximations are derived that are valid for angles on, and close to, the axial caustic associated with the glory.

4
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.

5
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.

6
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.

7
The PSA demonstrates that glory structure arises from nearside–farside (NF) interference, in an analogous way to the two-slit experiment.

8
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.

9
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.

10
In addition, various subsidiary approximations are derived.

11
The input to the theory consists of accurate quantum scattering matrix elements.

12
The theory also has the important attribute that it provides physical insight by bringing out semiclassical and NF aspects of the scattering.

13
The theory is used to show that the enhanced small angle scattering in the F + H2(vi = 0, ji = 0, mi = 0) → FH(vf = 3, jf = 3, mf = 0) + H reaction is a forward glory, where vi, ji, mi and vf, jf, mf are initial and final vibrational, rotational and helicity quantum numbers respectively.

14
The forward angle scattering for the H + D2(vi = 0, ji = 0, mi = 0) → HD(vf = 3, jf = 0, mf = 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.

Introduction

15
The differential cross section for a state-to-state reactive molecular collision often exhibits a complicated interference pattern.

16
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.

17
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

18
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.

19
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.

20
NF theory has been developed for PWS expanded in a basis set of Legendre polynomials, associated Legendre functions or reduced rotation matrix elements.

21
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

22
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°.

23
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

24
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

25
The scattering pattern is again interpreted as the interference between a small number of simpler (semiclassical) subamplitudes.2,29–31,33–38

26
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.

27
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

28
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

29
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 θR, where θR is the reactive scattering angle.

30
The factor 1/sin θR is characteristic of an axial caustic,39–43 which suggests that the enhanced intensity at forward angles is an example of glory scattering.

31
Unfortunately, the semiclassical CAM theory presented in refs. 29 and 30 is not uniformly valid at θR = 0.

32
Sokolovski has partially overcome this defect in a letter whose main emphasis is the relation between CAM pole structure and forward angle scattering.38

33
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 θR.

34
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:

35
(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.

36
A further approximation to the STA yields the glory formula used by Sokolovski.38

37
Transitional approximations are valid for angles on, and close to, the axial caustic, i.e., for θR ≈ 0, but they become increasingly inaccurate at larger angles.

38
(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 θR = 0.

39
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.

40
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.

41
(3) A uniform semiclassical approximation (USA) is derived which correctly interpolates between θR ≈ 0, where the transitional approximations are valid, and larger angles where the PSA is valid.

42
The derivation of the USA makes use of mathematical techniques for the (uniform) asymptotic evaluation of oscillating integrals.45–51

43
The ITA, STA, PSA, CSA and USA, together with some subsidiary approximations, are derived in section II.

44
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).

45
The theory of section II is applied in sections III and IV to two benchmark state-to-state reactions that exhibit forward angle scattering.

46
These are the F + H2 → FH + H and H + D2 → HD + D reactions respectively.

47
Section V contains our conclusions.

48
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.

Theory of forward glory scattering for reactive molecular collisions

Introduction

49
We consider a state-to-state chemical reaction of the type A + BC(vi, ji, mi = 0) → AB(vf,jf,mf = 0) + Cwhere vi, ji, mi and vf, jf, mf are the vibrational, rotational and helicity quantum numbers for the initial and final states, respectively.

50
Because the helicity quantum numbers are equal to zero, the scattering amplitude, f(θR), 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, SJ is a scattering matrix element, PJ(•) is a Legendre polynomial of degree J and θR ∈ [0,π] is the reactive scattering angle, i.e., the angle between the outgoing AB molecule and the incoming A atom.

51
The subscript, vi, ji, mi → vf, jf, mf, has been omitted from f(θR) and SJ for notational simplicity, as has the subscript, vi, ji, from k.

52
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).

53
The differential cross section is given by σ(θR) = |f(θR)|2.For the theoretical development that follows, it is convenient to first write the PWS (1) in the standard form with θR ∈ [0,π], where the argument of the Legendre polynomial is cos θR rather than cos(π – θR).

54
The modified scattering matrix element J in eqn. (3) is defined by J = exp(iπJ)SJ, J = 0,1,2,…

55
We begin the derivation by applying the Poisson summation formula (e.g., ref. 39) to the PWS (3), which yields the exact result for θR ∈ [0,π) where λ = J + 1/2 and the notation J = λ–1/2 ≡ (λ)has been used.

56
In eqn. (4), (λ) and Pλ–1/2(cos θR) have been interpolated (and extrapolated where necessary) from half-integer values of λ = 1/2,3/2,5/2,… to continuous values, λ ∈ [0,∞).

57
Sections III and IV describe the interpolation used for (λ).

58
In practice, separate interpolations are used for |(λ)| and arg (λ), where arg does not in general denote the principal value, rather arg (λ) is a continuous function with arg (λ) ∈ (–∞,∞).

59
In eqn. (4), Pλ–1/2(cos θR) is a Legendre function of the first kind of real degree, λ – 1/2, with θR ∈ [0,π).

60
[n.

61
b., Pλ–1/2(cos θR) is singular at θR = π, except for λ = 1/2,3/2,5/2,…]

62
Under semiclassical conditions, kR ≫ 1, where R is the reaction radius, the integrands in the series (4) are highly oscillatory functions.

63
We assume, as is commonly the case,39–41 that any points of stationary phase occur only for the m = 0 integrand.

64
We then have for θR ∈ [0,π)The upper limit, λmax + 1/2 = Jmax + 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 J ≡ 0 for J > Jmax.

65
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).

66
Eqn. (6) is the starting point for the approximations derived below.

67
In the derivations, we shall often expand arg (λ) about a real point λ = λs in a Taylor's series to second order arg (λ) = arg (λs) + (λs) (λ – λs) + ½′(λs) (λ – λs)2wherewill be called the quantum deflection function.

68
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

69
The adjective “quantum” indicates that {J}, and its continuation (λ), are in principle exact quantum quantities (although, if convenient, approximate scattering matrix elements could be used).

70
In addition, the notations andhave been employed in the Taylor expansion (7).

71
Figs. 1(a) and 1(b) show, for the F + H2 reaction of section III, the behaviour of arg (λ) and (λ) around the point λ = λg where arg (λ) is stationary (a maximum) so that (λg) = 0.

72
In addition, Fig. 1(b) illustrates the quantities λ±(θR) and λ(θR;φ) with φ ∈ [0,π] which play an important rôle in the derivations below.

Transitional approximations (TA)

73
In this section, two transitional approximations are derived for the forward glory scattering.

74
A TA is valid on, and close to, the axial caustic, i.e., for values of θR close to θR = 0.

Integral transitional approximation (ITA)

75
When θR = 0, we have Pλ–1/2(1) = 1 for all λ ∈ [0,∞).

76
For θR close to θR = 0, we assume that Pλ–1/2(cos θR) is slowly varying relative to the oscillations of (λ) and can be removed from the integrand of eqn. (6).

77
The (real) value used for λ in the removal of Pλ–1/2(cos θR) is that given by the stationary phase condition (λg) = 0which defines the glory angular momentum variable, λg, see Fig. 1.

78
In practice, there may be more than one root to eqn. (9)-see sections III and IV for examples.

79
In this situation, we use the λg that makes the largest contribution to the integral in eqn. (6), when the stationary phase approximation is employed (see section IIB.2).

80
We can now approximate eqn. (6) by fITA(θR) = GPλg–1/2(cos θR)where G denotes the complex-valued integralThe corresponding differential cross section isσITA(θR) = |G|2Pλg–1/2(cos θR)2Eqns. (10)–(12) define the ITA for forward glory scattering.

81
Note that the value of the cross section at θR = 0 is finite, being σITA(θR = 0) = |G|2.

Semiclassical transitional approximation (STA)

82
The ITA can be further approximated if we evaluate the integral (11) by the stationary phase method.

83
We again assume that the stationary phase condition is given by eqn. (9) and that the situation in Fig. 1 applies.

84
Then with the help of the stationary phase result [ref. 55, eqns. (4.1) and (4.12)] where F′(λs) = 0, a < λs < b, G(λs) ≠ 0,∞ and F″(λs) ≠ 0,∞, we obtain for the STA The corresponding differential cross section is.

85
Eqns. (14) and (15) define the STA for forward glory scattering.

86
At θR = 0, the value of the cross section is Note that eqn. (7) becomes for this case arg (λ) = arg (λg) + ½′(λg) (λ – λg)2where ′(λg) is negative, i.e., arg (λ) has a maximum at λ = λg as illustrated in Fig. 1.

Remarks

87
 

88
(i) The ITA is a global approximation, in that the input consists of {J}, and its continuation (λ), which contain all the information on the dynamics of the reaction.

89
In contrast, the STA is a local approximation, since it only requires dynamical information at λ = λg.

90
(ii) Both the ITA and STA predict that the angular distribution for forward glory scattering is proportional to Pλg–1/2(cos θR)2.

91
However conversely, if we find that the angular scattering at θR ≈ 0 can be fitted to the form constant × Pλt–1/2(cos θR)2 say, then it does not follow that we have a forward glory without further investigation, e.g., we must check that (λt) = 0 is satisfied.

92
For there may be other dynamical mechanisms that result in the integral (11) being dominated by a small range of λ values around λ = λt, but with the property, (λt) ≠ 0.

93
(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.

94
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].

95
The appendix examines five ways in which a Legendre function can be approximated by a Bessel function of order zero.

96
The Hilb approximation is the one recommended for use: which has an error, O(λ–3/2), that is uniform with respect to θR ∈ [0,π – ε] with ε > 0.

97
At θR = 0, where eqn. (17) takes the indeterminate form 0/0, we use the well-known limit, θR/sinθR → 1 as θR → 0.

98
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.

99
If we make the additional approximation of replacing the factor (θR/sinθR)1/2 by unity for θR ≈ 0 in the STA/J0, we obtain the forward glory approximation used by Sokolovski-see eqn. (17) of .ref. 38

100
(iv) The ITA and STA both predict that the cross section is zero at the roots of the equation, Pλg–1/2(cos θR) = 0.

101
Similarly the ITA/J0 and STA/J0 predict zero cross sections at the roots of J0(λgθR) = 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).

Primitive semiclassical approximation (PSA)

102
As θR moves away from θR = 0, the assumption of the ITA and STA that Pλ–1/2(cos θR) is slowly varying becomes less valid.

103
Instead, we must use the asymptotic approximation which is valid for λ sinθR ≫ 1.

104
The two exponentials in eqn. (18) have a physical interpretation: The term exp(+iλθR) describes an angular wave travelling anti-clockwise in θR and corresponds to farside (F) scattering in a semiclassical treatment, whilst the term exp(–iλθR) represents an angular wave travelling clockwise in θR and corresponds to nearside (N) scattering.14–30

105
Substituting approximation (18) into the integral (5) lets us write f(θR) = f+(θR) + f(θR) where the F and N subamplitudes are with

106
Next we apply the stationary phase approximation to the I±(θR) integrals, assuming that |(λ)| is slowly varying.

107
The stationary phase condition is or (λ) = ∓θR for λ = λ±(θR) where we have assumed that there exist a unique pair of real valued roots to eqn. (23).

108
The λ±(θR) are illustrated in Fig. 1(b).

109
Note that λ(θR) and λ+(θR) are coincident at θR = 0, since λg = λ±(θR = 0).

110
Also Fig. 1(b) shows that ′(λ±(θR)) < 0.

111
Explicit formulae for the λ±(θR) can be obtained when the quadratic approximation (16) is made for arg (λ).

112
For (λ), we have (λ) = –|′(λg)|(λ – λg) and the stationary phase eqn. (23) then gives

113
Applying the stationary phase approximation (13) to the integrals (21) yields The term exp[+iλ±(θR)θR] in eqn. (25) describes an angular wave travelling anti-clockwise in θR, so that f+(θR) corresponds to the F scattering, whilst the term exp[–iλ±(θR)θR] represents an angular wave travelling clockwise in θR, so that f(θR) corresponds to the N scattering.

114
The formulae for the f±(θR) subamplitudes take a simpler form if we first define the F and N cross sections The σ±(θR) are analogous to classical cross sections.

115
This analogy can be made more explicit by defining the impact parameter variable, b = λ/k together with b±(θR) = λ±(θR)/k.

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+(θR) = –[σ+(θR)]1/2exp[iβ+(θR)]andf(θR) = –i[σ(θR)]1/2exp[iβ(θR)]where the F and N phases β±(θR) are defined byβ±(θR) = arg (λ±(θR)) ± λ±(θR)θR.

117
The full PSA scattering amplitude is then given by fPSA(θR) = –[σ+(θR)]1/2 exp[iβ+(θR)] – i[σ(θR)]1/2 exp[iβ(θR)].

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).

119
Eqns. (26)–(30) define the PSA for the scattering amplitude.

120
The corresponding PSA differential cross section for glory scattering is given by σPSA(θR) = σ+(θR) + σ(θR) + 2[σ+(θR)σ(θR)]1/2sin[β+(θR) – β(θR)].

121
Eqn. (31) has the typical form of a two slit interference pattern for the F and N scattering.

Classical semiclassical approximation (CSA)

122
If the sinusoidal term in eqn. (31) is neglected, we obtain a result that will be called the CSA σCSA(θR) = σ+(θR) + σ(θR)σCSA(θR) can be estimated by quasiclassical trajectory (QCT) calculations.

123
If the |(λ±(θR))|2 in eqn. (26) contain a sizable contribution from tunnelling, then we might expect the QCT method to underestimate σCSA(θR).

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 σCSA(θR).

125
It is interesting to take the θR → 0 limit of eqn. (32).

126
We find The singularity at θR = 0 can be clearly seen in eqn. (33).

127
Its origin is the (non-uniform) asymptotic approximation (18), which is not valid for θR → 0.

128
Eqn. (33) is analogous to the classical result for elastic scattering from a central potential.

129
[n.

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].

131
We can also derive eqn. (33) from the STA intensity (15), by averaging over the oscillations in Pλg–1/2(cos θR)2.

132
With the help of eqn. (18), we find 〈Pλg–1/2(cos θR)2〉 ≈ 1/(πλgsin θR) so that eqn. (15) reduces to eqn. (33).

133
An alternative derivation of eqn. (33) uses the Hilb approximation (17) together with the average 〈J0(λgθR)2〉 ≈ 1/(πλgθR).

Uniform semiclassical approximation (USA)

Introduction

134
The ITA and STA do not merge smoothly with the PSA when θR increases from θR = 0 to larger angles.

135
We can rectify this defect by deriving a USA which does change smoothly as θR moves away from the axial caustic and which contains the ITA/J0 and PSA as limiting cases.

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).

137
The derivation of the USA proceeds by applying uniform asymptotic techniques to a two-dimensional integral representation for f(θR).

138
The first step in the derivation inserts the Hilb approximation (17) for Pλ–1/2(cos θR) 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)].

139
In the second step, we insert eqn. (35) into eqn. (34) obtaining the following two-dimensional integral for f(θR) where the phase is defined by B(θR;λ,φ) = arg (λ) – λθRcosφ.

140
Before proceeding, it is first helpful to examine the stationary points of B(θR;λ,φ).

Stationary points of B(θR;λ,φ)

141
The stationary phase equations for the two dimensional integral (36) are where eqn. (8) has been used.

142
For θR ≠ 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.

143
Substituting the root φ = 0 into eqn. (38) gives (λ(θR)) = +θR whilst substituting the root φ = π into eqn. (38) leads to (λ+(θR)) = –θR, which agree with eqn. (23).

144
The stationary points of B(θR;λ,φ) are therefore (λ(θR),φ = 0) and (λ+(θR),φ = π).

145
The values of B(θR;λ,φ) at the stationary points are B(θR;λ±(θR),φ = π,0) = arg (λ±(θR)) ± λ±(θR)θR ≡ β±(θR)where the definition of the β±(θR) in eqn. (29) has been used.

146
It is convenient to continue to assume θR ≠ 0 in the following calculations of stationary phase points.

Stationary phase integration over λ

147
The phase B(θR;λ,φ), defined by eqn. (37), is a rapidly varying function of λ.

148
It is therefore legitimate to apply the stationary phase approximation to the integral over λ in eqn. (36).

149
There is one stationary point, λ = λ(θR;φ), given by Note that λ(θR;φ) varies from λ(θR;φ = 0) ≡ λ(θR) to λ(θR;φ = π) ≡ λ+(θR) as φ changes from 0 to π, as illustrated in Fig. 1(b).

150
Applying the stationary phase formula (13) to eqn. (36), and noting that together with ′(λ(θR;φ)) < 0 [see Fig. 1(b)], we obtain Remark:

151
When θR ≈ 0, the phase B(θR;λ(θR;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.

152
However, as θR moves to larger angles, B(θR;λ(θR;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).

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].

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 < λs < b.

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.

156
[ref. 55, p. 77].

157
We then find that the stationary phase evaluation of the integral (42) results in the PSA, eqn. (30) for f(θR).

Exact change of variable

158
When θR ≈ 0, the phase of the integrand in eqn. (42), namely B(θR;λ(θR;φ),φ) = arg (λ(θR;φ)) – λ(θR;φ)θRcosφwould be the same as the phase of the Bessel function of order zero, eqn. (35), were it not for the slow dependence of λ(θR;φ) on φ.

159
This suggests that we make the following exact one-to-one change of variableB(θR;λ(θR;φ),φ) = A(θR) – ζ(θR)cosψ where the new parameters A(θR) and ζ(θR) are functions of the old parameter θR, whilst the new variable ψ = ψ(θR;φ) is a function of both θR and the old variable φ [and inversely, φ = φ(θR;ψ)].

160
The rhs of eqn. (45) can also be regarded as the unfolding of a germ of infinite codimension.

161
The stationary phase condition for the lhs of eqn. (45) has already been written down in eqns. (43) and (44).

162
The two stationary points are φ = 0 and φ = π for φ ∈ [0,π].

163
For the rhs of eqn. (45), the stationary phase condition is ζ(θR)sinψ = 0 and the two stationary points are ψ = 0 and ψ = π for ψ ∈ [0,π].

164
The requirement that the transformation (45) be one-to-one, and hence that dφ/dψ ≠ 0,∞, is used to determine A(θR) and ζ(θR).

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.

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.

167
This is done in section IID.6 below.

168
Inserting the correspondence φ = 0 ↔ ψ = 0 into eqn. (45) gives β(θR) = A(θR) – ζ(θR) where eqn. (40) has been used.

169
Similarly, inserting the correspondence φ = π ↔ ψ = π into eqn. (45) yields β+(θR) = A(θR) + ζ(θR).

170
Solving the linear eqns. (48) and (49) results in A(θR) = ½[β+(θR) + β(θR)] and ζ(θR) = ½[β+(θR) – β(θR)].

171
Eqns. (50) and (51) are the sought-for relations which express A(θR) and ζ(θR) in terms of the F and N phases, β±(θR).

172
Next we use eqn. (45) and the correspondences (47) to change variable from φ to ψ in the integral (42) for f(θR).

173
We obtain where φ is to be regarded as a function of ψ, i.e., φ = φ(θR;ψ).

Approximation of the pre-exponential factor

174
Next we express the pre-exponential factor in eqn. (52) in the form{•} = p(θR) + q(θR)cosψ + r(θR;ψ)sinψinvolving the functions p(θR), q(θR) and r(θR;ψ).

175
When θR ≈ 0, the factor {•} is approximately constant since λ(θR ≈ 0;φ) ≈ λg [see Fig. 1(b)], whereas when θR 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.

176
These considerations suggest that we neglect the term r(θR;ψ)sinψ in eqn. (53) and approximate the pre-exponential factor by with φ = φ(θR;ψ).

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

Determination of p(θR) and q(θR)

178
To determine p(θR) and q(θR), we substitute the correspondences φ = 0 ↔ ψ = 0 and then φ = π ↔ ψ = π into eqn. (54), obtaining andwhere the relations λ(θR;φ = 0) ≡ λ(θR) and λ(θR;φ = π) ≡ λ+(θR) have also been used.

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.

180
To do this we rewrite eqn. (46) in the form and differentiate with respect to ψ, obtaining.

181
Substituting the correspondences φ = 0 ↔ ψ = 0 and φ = π ↔ ψ = π into eqn. (60), then gives and where the results λ(θR;φ = 0) ≡ λ(θR) and λ(θR;φ = π) ≡ λ+(θR) have again been used.

182
Substituting eqns. (61) and (62) into eqns. (58) and (59) and using eqns. (26), lets us write p(θR) and q(θR) in the form and

Uniform approximation

183
The USA scattering amplitude, fUSA(θR), is obtained upon combining eqns. (55), (63), (64) where A(θR) and ζ(θR) are defined by eqns. (50) and (51) respectively.

184
The corresponding USA differential cross section is given by.

185
Eqns. (65) and (66) express the USA scattering amplitude and cross section in terms of the F and N cross sections, σ±(θR), and phases, β±(θR).

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).

187
This implies that σUSA(θR), in contrast to the ITA and STA, is never equal to zero unless at some θR, we have σ+(θR) = σ(θR) and J0(ζ(θR)) = 0.

Limiting cases

188
We must also check that eqn. (65) contains as limiting cases the non-uniform PSA and STA/J0 scattering amplitudes.

189
The PSA is valid when θR moves to larger angles, where we have ζ(θR) ≫ 1.

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).

191
Next we consider the STA/J0, which is valid for θR → 0.

192
We make the following three approximations to deduce the limiting form of eqn. (65) when θR → 0:

193
(a) Since λ±(θR) → λg [see, Fig. 1(b)] and therefore, σ+(θR) → σ(θR) when θR → 0, we can neglect the term containing J1(ζ(θR)) in eqn. (65).

194
[N. b., we also have ζ(θR) → 0 for the argument of the Bessel function].

195
(b) For A(θR), we similarly find that A(θR) → arg (λg) as θR → 0.

196
(c) We also require the leading term of ζ(θR) as θR → 0.

197
We can find it using the quadratic approximation (16) for arg (λ) together with the definitions (29) and (51) for β±(θR) and ζ(θR) respectively.

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±(θR) = arg (λg) + ½θ2R/|′(λg)| ± λgθRwhich shows that ζ(θR) → λgθR as θR → 0.

199
Substituting these three limits into eqn. (65), we obtain which is the scattering amplitude for the STA/J0, see eqns. (14) and (17).

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.

201
In particular, the forward angle scattering associated with the FH(vf = 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)

202
In this section we apply the glory theory of section II to the angular scattering of the state-to-state transition F + H2(vi = 0, ji = 0, mi = 0) → FH(vf = 3, jf = 3, mf = 0) + H

Input scattering matrix data

203
The input to the glory analysis consists of a set of accurate quantum scattering matrix elements, {J} for J = 0(1)23 (and {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

204
The translational energy is 0.119 eV, which corresponds to a translational wavenumber of k = 10.2 Å–1.

205
Fig. 2(a) shows the modulus of the scattering matrix input data, |λ–1/2|, for λ = ½,,,…,23½ (solid circles), plotted versusλ, together with its real valued continuation, |(λ)|, drawn as a solid line.

206
The continuation is achieved by interpolating the discrete data (and extrapolating to λ = 0) with polynomials of degree 6.

207
It can be seen that |(λ)| possesses two peaks at λ ≈ 10 and λ ≈ 16, and a minimum at λ ≈ 135.

208
Fig. 2(b) displays the analogous plot for the input phase data, arg λ–1/2versusλ (solid circles) for λ = ½,,,…,23½, as well as its continuation, arg (λ) versusλ (solid curve).

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 (λ)/rad ∈ (0,40).

210
The phase arg λ–1/2 is calculated by adding multiples of 2π to the phases computed as principal values, until two successive values, arg λ–1/2 and arg λ+1/2 differ by less than π.

211
The continuation again uses polynomials of degree 6.

212
It can be seen that arg (λ) has a maximum at λ ≈ 16.

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.

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

215
The NF decomposition for the Legendre PWS (3) is (for θR ≠ 0,π) f(θR) = fN(θR) + fF(θR) where the NF subamplitudes are defined by in which QJ(cos θR) is a Legendre function of the second kind.

216
The corresponding NF angular distributions are given by σN,F(θR) = |fN,F(θR)|2.

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.

218
Fig. 3 shows the results of the NF analysis for the PWS scattering amplitude, eqn. (3).

219
It displays log plots of σ(θR), σN(θR) and σF(θR) versusθR.

220
At backward angles the scattering is N dominated, with the oscillations arising from NF interference.

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

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.

223
The change in mechanism for the forward and backward angle scattering occurs around θR = 40°, which implies that the glory theory of section II should only be used for θR ≲ 40° (and in particular, the accuracy of the USA glory equations is expected to degrade as θR increases from θR = 0° to θR ≈ 40°).

224
The σ(θR), σN(θR) and σF(θR) 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.

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

226
The genericity of the cross sections led Dobbyn et al22. to suggest that the same mechanism also occurs for the F + H2 reaction.

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

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

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.

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.

231
This confirms that kR ≫ 1, which is the condition for the semiclassical theory of section II to be valid.

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.

233
For θR = 0, at which angle Pλ–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).

234
More generally, the angular distribution computed from eqn. (6) by quadrature for θR ∈ [0°,90°] agrees with the PWS cross section, eqn. (2), to within graphical accuracy (for θR > 90°, integrals with m = ±1 in the Poisson series, eqn. (4), start to become important numerically).

235
These tests therefore justify the neglect of integrals with m ≠ 0 in the Poisson series, eqn. (4), at forward angles.

236
The quantum deflection function, (λ), defined by eqn. (8), plays a key role in the glory theory of section II.

237
Fig. 4 shows a plot of (λ) versusλ for the F + H2 reaction.

238
It can be seen that eqn. (9) has a (glory) root at λg ≈ 16.0 together with additional roots at λg ≈ 18.4,20.3 and 23.2.

239
However |(λ)|2 is very small at these additional roots, being ≈10–4, ≈10–6 and ≈10–8 respectively; their contribution has therefore been neglected.

240
The ITA, defined by eqns. (10)–(12), and the STA, defined by eqns. (14) and (15), require an accurate value for λg.

241
This was obtained by fitting a polynomial of degree 4 to the input phase data, arg λ–1/2, for λ = 13½(1)17½, as illustrated in Fig. 1(a).

242
We then find on differentiation and solving eqn. (9) that λg = 16.02.

243
We also find that ′(λg) = –0.522.

244
Fig. 5 compares the PWS cross section of eqn. (2) with the ITA angular distribution for the angular range, 0° ≤ θR ≤ 40°.

245
It can be seen there is close agreement between the PWS and ITA angular distributions for the forward peak up to θR ≈ 10°; then the ITA becomes less accurate as θR increases.

246
This loss of accuracy is expected since (a) the ITA is, by construction, a transitional approximation designed to be most reliable at θR = 0°, and (b) the change in mechanism discussed in section IIIB becomes increasingly important as θR moves towards θR = 40°.

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 (θR/sinθR)1/2 by unity (see also the discussion in the Appendix).

248
Fig. 5 also displays the glory scattering predicted by the USA.

249
It can be seen that the USA is more accurate than the ITA, in particular for the amplitudes of the subsidiary glory oscillations.

250
The change in mechanism at θR ≈ 40° is likely to account, in part, for the discrepancies between the USA and PWS angular distributions as θR increases.

251
The PSA agrees closely with the USA for θR ≳ 4°; it is not shown separately in Fig. 5.

252
Finally, the CSA is also plotted in Fig. 5.

253
As expected, it is monotonic and passes through the glory oscillations; the divergence of the CSA as θR → 0° can be clearly seen.

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.

255
In particular, a time-delayed mechanism for the forward angle scattering of the HD(vf = 3, jf = 0) final rovibrational state has been of much interest1,37,38,82,83 (for other recent research on this reaction, see .ref. 84–96)

256
The next section applies the glory theory of section II to the angular scattering of the state-to-state transition H + D2(vi = 0, ji = 0, mi = 0) → HD(vf = 3, jf = 0, mf = 0) + D

Input scattering matrix data

257
The input to the glory analysis consists of a set of accurate quantum scattering matrix elements, {J} for J = 0(1)30, with {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.

258
The corresponding translational energy is 1.81 eV, with the translational wavenumber being k = 26.4 Å–1.

259
The scattering matrix elements were computed by a new time dependent plane wavepacket method82,83,98–100 and were kindly supplied to the author by Dr S. C. Althorpe.

260
Fig. 6(a) shows the modulus input data, |λ–1/2|, for λ = ½,,,…,30½, plotted versusλ (solid circles), together with its real valued continuation, |(λ)| (solid curve).

261
Polynomials of degree three have been used to interpolate the discrete data (and to extrapolate to λ = 0).

262
It can be seen that |(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.

263
Thus the |(λ)| plot for the H + D2 reaction is more complicated than that for the F + H2 system in Fig. 2(a).

264
The plot for the input phase data, arg λ–1/2versusλ (solid circles) for λ = ½,,,…,30½, as well as its continuation, arg (λ) versusλ (solid curve), is shown in Fig. 6(b).

265
The continuation also uses polynomials of degree three.

266
It can be seen that arg (λ) has a maximum at λ ≈ 24.

267
The noticeable drop in arg λ–1/2 on going from λ = 18½ to λ = 19½ is associated with the corresponding near zero in |λ–1/2| evident in Fig. 6(a).

Nearside–farside analysis

268
Fig. 7 shows log plots of σ(θR), σN(θR) and σF(θR) versusθR.

269
At backward angles the scattering is N dominated, with the weak oscillations arising from NF interference.

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

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.

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.

273
At around θR = 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 θR increases towards 40°.

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°.

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

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(vi = 0, ji = 0), to the final state HD(vf = 3, jf = 0) + D.

277
The movie showed that the HD(vf = 3, jf = 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

278
Subsequently, I had private viewings of the movie, courtesy of Dr Althorpe.

279
I have also examined clips and stills from the movie.

280
The NF analysis in Fig. 7 is consistent with the two mechanism picture of the reaction dynamics depicted in the movie.

281
Thus N scattering of the reactants, which dominates backward angles, is clearly visible in the movie and corresponds to the direct recoil mechanism.

282
In contrast, the N and F scattering in the angular range, θR ≲ 40°, which arise from decaying Regge states (surface waves),37,38 correspond to the indirect time-delayed mechanism.

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.

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.

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).

286
Note that one difference is that H + D2 is N dominated for θR ≲ 40°, whereas the F + H2 reaction is F dominated in this angular range.

Glory analysis of forward scattering

287
The H + D2 reaction has a radius of interaction of approximately R = 2 Å.

288
Thus kR ≈ 50, which means that the semiclassical theory of section II is applicable.

289
We first examine the validity of retaining just the m = 0 integral in the Poisson series (4).

290
For θR = 0, where Pλ–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.

291
In fact, the angular distribution computed by quadrature from eqn. (6) agrees with the PWS cross section (2) to within graphical accuracy for θR ∈ [0°,90°].

292
As for the F + H2 reaction, integrals with m = ±1 in the Poisson series (4) only become important numerically when θR > 90°.

293
Thus, at forward angles, we can safely neglect integrals with m ≠ 0 in the Poisson series (4).

294
Fig. 8 shows a plot of (λ) versusλ for the H + D2 reaction.

295
The plot is more irregular than the one for the F + H2 example of Fig. 4.

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.

297
Fig. 8 shows that eqn. (9) has a (glory) root at λg ≈ 24 together with additional roots at λg ≈ 18.5,19.6 and 29.7.

298
The first two roots are associated with the near-zero of (λ) at λ ≈ 19.0.

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.

300
The root near λ = 29.7 has |(λg)|2 ≈ 10–8; its contribution has also been neglected.

301
Next we apply the glory approximations derived in Section II.

302
We recall that the ITA requires a value for λg, the STA requires values for λg and the slope of (λ) at λ = λg, whilst the USA, PSA and CSA require values for λ±(θR) and the slopes of (λ) at λ = λ±(θR) for θR ≲ 40°.

303
The irregularities in the (λ) curve of Fig. 8 near λ ≈ 24 make it difficult to extract accurate values for these quantities.

304
To avoid this difficulty, we interpolated the input phase data, arg λ–1/2 for λ = 21½(1)25½ to determine λg, and then approximated (λ) by a straight line—in effect the input data has been smoothed.

305
This procedure resulted in λg = 24.19 and ′(λg) = –0.233.

306
These values are very close to those obtained by Sokolovski, namely λg = 24.2 and ′(λg) = –0.24, using a smoothed Padé reconstruction.

307
The straight-line approximation for (λ) is drawn as a dashed line in Fig. 8 for (λ)/deg ∈ [–40,40].

308
Values for the λ±(θR) needed in the USA, PSA and CSA are then given by eqn. (24).

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° ≤ θR ≤ 10° and 10° ≤ θR ≤ 40° (two angular ranges are used in order to show more clearly the errors of the different approximations).

310
There is close agreement between the PWS and ITA angular distributions for the forward peak up to θR ≈ 12°, with the ITA then becoming less accurate as θR increases.

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 θR = 0°, and (b) the change in mechanism discussed in sect. IVB becomes increasingly significant as θR → 40°.

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 (θR/sinθR)1/2 replaced by unity (see the Appendix for some additional discussion).

313
Fig. 9 also displays the glory scattering predicted by the USA.

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.

315
The change in mechanism at θR ≈ 40° is likely to account, in part, for the discrepancies between the USA and PWS angular distributions as θR increases.

316
The PSA is not shown separately in Fig. 9 since it agrees closely with the USA for θR ≳ 3°.

317
The cross section for the CSA in Fig. 9 is seen to be monotonic.

318
It passes through the glory oscillations and is divergent as θR → 0°.

319
Finally, we note that Althorpe has introduced82,83,99,100 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.

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.

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.

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.

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.

324
The PSA clearly demonstrates that glory structure arises from nearside–farside (NF) interference, in an analogous way to the two-slit experiment.

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.

326
Rather, the input to the theory consists of accurate quantum scattering matrix elements.

327
The theory has been used to show that the enhanced small angle scattering in the F + H2(vi = 0, ji = 0, mi = 0) → FH(vf = 3, jf = 3, mf = 0) + H reaction is a forward glory.

328
And the same is also true for the H + D2(vi = 0, ji = 0) → HD(vf = 3, jf = 0) + D reaction, in agreement with a simpler treatment by Sokolovski38 which is a special case of the present theory.