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Semiclassical angular scattering in the F+H2→HF+H reaction: Regge pole analysis using the Padé approximation
5 November 1999
Chemical Physics Letters 313 Ž1999. 225–234 www.elsevier.nlrlocatercplett
Semiclassical angular scattering in the F q H 2 ™ HF q H reaction: Regge pole analysis using the Pade´ approximation D. Sokolovski a
a,)
, J.F. Castillo b, C. Tully
a
Theoretical and Computational Physics Research DiÕision, Department of Applied Mathematics and Theoretical Physics, Queen’s UniÕersity of Belfast, Belfast BT7 1NN, UK b Departamento de Quimica Fisica I, UniÕersidad Complutense de Madrid, Madrid 28040, Spain Received 22 March 1999; in final form 24 May 1999
Abstract We apply the complex angular momentum analysis to study state-selective forward scattering in the F q H 2 reaction. The scattering matrix calculated by Castillo et al. is used. The S-matrix element is analytically continued into the complex plane of the total angular momentum with the help of the Pade´ approximation. Resonance ŽRegge. poles are identified and their residues evaluated. The semiclassical optical model is corrected to account for finite lifeangles of leading resonances. For chosen transitions, the forward scattering enhancement is shown to be a resonance effect due to the decay of a Regge state with angular life of about 328. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Reactive angular distributions in the F q H 2 ™ FH q H reaction have been a subject of much attention for more than a decade w1–5x. Their main feature, the state-selective forward angle scattering, was observed by Neumark and co-workers in 1985 w1x. Neumark et al. suggested that the enhanced forward scattering for HF Ž ÕX s 3. was due to the existence of a long-lived state formed during the reaction w1x. However, quasiclassical trajectory studies on the Stark–Werner ŽSW. potential energy surface ŽPES. w2x have demonstrated that classical trajectories producing HF Ž ÕX s 3. also have a tendency to be scattered in the forward direction, thus suggesting a different explanation for the effect. Quantum-mecha) Corresponding author. Fax: q44-1232-239-182; e-mail: [email protected]
nical calculations on the SW PES w3,4x showed even greater forward scattering. Whether enhanced forward scattering in F q H 2 has a direct or resonance origin remained an open question. The authors of Ref. w3x analysed time delays associated with the peak structure in the cumulative reaction probability at zero total angular momentum J versus energy E and concluded that no long-lived state was involved. An additional observation was made by Dobbyn et al. w5x. They have pointed out that, for a light system, the angular life of a resonance can be considerable even if the complex is relatively short-lived. It was also demonstrated that the direct scattering optical model of Herschbach w6x fails to reproduce correctly the small angle scattering for F q H 2 ™ FH q H. Using a more sophisticated nearside–farside analysis w6x the authors of Ref. w5x suggested that the oscillatory structures in the angular distributions were consistent with the presence of
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 0 1 6 - 7
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
226
a single Regge state with a lifeangle of 308–408 at a total energy E s 0.4309 eV. A more rigorous analysis of the problem is required w5x. In this Letter, we present a detailed analysis of resonance contributions to the zero helicity state-tostate differential cross-sections obtained in quantum calculations w4x. In a slightly modified form, we shall apply the semiclassical theory of resonance angular scattering developed in Refs. w7,8x. This Letter is organised as follows: in Section 2, we apply the Poisson sum formula to obtain an integral representation for the reactive scattering amplitude. In Section 3, we use the Pade´ approximation to analytically continue the S-matrix elements into the complex plane of the total angular momentum and identify the resonance poles. In Section 4, we use the Pade´ approximation to construct a semiclassical optical model similar to that of Herschbach w6x. This model will fail if resonances with large lifeangles are present. In Section 5, we define the lifetime correction function ŽLCF. to correct the optical model. In Section 6, we give the results for the F q H 2 ™ FH q H reactions. Section 7 contains our conclusions. 2. Reactive scattering amplitude For an atom–diatom collision, A q BC ™ AB q C the reactive scattering amplitude fn X § n is given by the partial wave series w9,10x, fn X § n Ž u .
Fig. 1. Initial and final orientations of the inelastic Jacobi vector R BC ™ A and the final direction of the reactive Jacobi vector RAB ™ C . The total angular momentum J is perpendicular to the plane of the drawing. In the course of collision R BC ™ A sweeps the angle w Žshaded.. Another final direction of RAB ™ C , leading to the same scattering angle u Ž w sp q u . is shown by a dashed line. Also shown is the reactive scattering angle u R .
Using the semiclassical asymptote of PJ Žcos u . for large J w12x PJ Ž cos u . f
`
s Ž ikn .
y1
Ý Ž J q 1r2.
d MJ X M
Žu .
SnJX § n
,
JsJmin
Ž 1.1 . where n , Ž n X . is a collective index, n ' Ž Õ, j, M ., comprising the vibrational and rotational quantum numbers of the molecule, Õ and j, and the momentum Ž p-. helicity M Žprojection of the molecule’s angular momentum onto the atom–molecule relative velocity.. In addition, J is the total angular momentum quantum number, u is the centre of mass scattering angle ŽFig. 1., kn is the translational wavevector in the incident channel and Jmin ' maxŽ< M <, < M X <.. If both helicities are zero, M s M X s 0, the Wigner matrices d MJ X M Ž u . w11x reduce to the Legendre polynomials PJ , J d 00 Ž u . s PJ Ž cos u . . Ž 1.2 .
1r2
2
Ž J q 1r2. p sin u = cos Ž J q 1r2 . u y pr4 ,
Ž 1.3 .
1rJ - u - p y 1rJ . and applying the Poisson sum formula w12x to Eqs. Ž1.1. and Ž1.2. we can rewrite the scattering amplitude fn X § n Ž u . in an equivalent form fn X § n Ž u . f Ž ik i .
y1
Ž 2 p sin u .
y1r2
`
Ý f˜n § n Ž w k . exp Ž yipr4 y ik pr2.
=
X
ksy`
Ž 1.4 . where f˜n X § n Ž w . '
`
H0 exp Ž i lw . S
X
Õ §Õ
Ž l . l1r2 d l ,
y `- w-` , Sn X § n Ž J q 1r2 . s SnJX § n ,
Ž 1.5 .
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
and k
½
w k Ž u . s Ž y1 . u q p k q 1 y Ž y1 . p k - w k - p Ž k q 1. .
k
5
r2 ,
Ž 1.6 .
We shall use representation Ž1.4. – Ž1.6. to study resonance effects in reactive angular distributions. Semiclassically, the variable w in Eq. Ž1.5. coincides w13x with the angle by which the Jacobi vector R BC ™ A Ži.e. the vector connecting the centre of mass of BC with A. rotates during the collision around the fixed total angular momentum J ŽFig. 1. and f˜n X § n Ž w . is proportional to the probability amplitude for R BC ™ A to rotate by w . How far R BC ™ A will precess around J depends on the reaction mechanism. If the forces between collision partners are mostly repulsive, direct reactive classical trajectories have 0 - w - p and f˜n X § n Ž w . is likely to be contained between w s 0 and w s p. If, on the other hand, the atoms form a long-lived intermediate state, rotation of the decaying complex Žrequired to maintain the total angular momentum J . may carry R BC ™ A to larger w s, and f˜n X § n Ž w . would extend beyond the w0,p x region. Since the full scattering amplitude Ž1.4. is obtained by adding, with appropriate phases, contributions from all rotation angles consistent with the scattering angle u Žthese are given by Eq. Ž1.6.. we expect resonances with sufficiently large lifeangles lead to observable interference effects in reactive angular distributions w7,8x. Next we shall use Eqs. Ž1.4., Ž1.5. and Ž1.6. to investigate the origin of forward scattering of the zero-helicity Ž M s M X s 0. state-to-state angular distributions of the F q H 2 ™ FH q H reaction observed in calculations w3x.
3. Pade´ approximation for the S-matrix. Resonance Regge poles Usually, a compute code evaluates the S-matrix elements SnJX § n , for integer values of J, J s 0,1,2,3 . . . N. To reconstruct from this discrete set of values an analytic function Sn X § n X Ž l. in Eq. Ž1.5., we shall use the Pade´ approximation w14,15x. We note first that, for a heavy system, the S-matrix element is likely to have a phase which rapidly
227
varies with l. To avoid reproducing these rapid oscillations we write Sn X § n Ž l . s exp i f Ž l . sn X § n Ž l . Ž 2.1 . Ž . where f l is a known analytical function chosen so that remaining sn X § n Ž l. varies with l sufficiently slowly. For sn X § n X Ž l. we have a Pade´ approximation d ri Pdy1 Ž l . sn X § n Ž l . s Ý ' Ž 2.2 . Q d Ž l. is1 l y l i where Pdy 1Ž l. and Q d Ž l. are polynomials of l of order d y 1 and d, respectively. A detailed description of the Pade´ method was given by Bessis et al. in Ref. w15x and, to avoid repetition, we shall only review the procedure briefly. Choosing 2 d half-integer values lŽ k ., k s 1,2, . . . ,2 d ( N and using Eqs. Ž2.1. and Ž2.2. yields 2 d equations for 2 d unknowns r i and l i , i s 1,d d ri S ÕX § Õ Ž lŽ k . . exp yi f Ž lŽ k . . s Ý Ž k . , is1 l y l i k s 1,2, . . . 2 d . Ž 2.3 . The non-linear Eq. Ž2.2. can be solved by expressing the function in the r.h.s as a continued fraction w14x. This allows for determination of the coefficients of the polynomials Pdy 1Ž l. and Q d Ž l. in Eq. Ž2.2. and, finally, of the quantities r i and l i . Note that a successful Pade´ approximant should not have any sharp features between the grid points lŽ k .. Also, all physical resonance poles should lie in the first quadrant of the complex l-plane. Under these conditions Eqs. Ž2.1. and Ž2.2. provide an analytic continuation of the S-matrix element in the vicinity of the real l-axis. Poles of the S-matrix close to the real axis correspond to metastable states of the triatomic w7,8x. Separating in Eq. Ž2.3. those, say, Nres Ž Nres - d . poles which correspond to the resonances from those which contribute to the smooth ‘background’ q Ž l., we have Sn X § n Ž l . s exp i f Ž l .
Nres
rm
ms1
l y lm
½Ý
q q Ž l.
5
Ž 2.4 . where q Ž l. includes all remaining terms in Eq. Ž2.3.. The resonance poles in Eq. Ž2.4. may be chosen to ensure that the remaining q Ž l. has as little structure as possible.
228
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
Fig. 2. Ža. Differential cross-sections for the Ž3,2,0. § Ž0,0,0. transition at E s 0.3872 eV vs. u R s p y u : exact PWS Eq. Ž1.1. Žsolid.; optical model Eq. Ž3.5. Žshort dash.; corrected optical model Eq. Ž4.9. with only m s 1 pole in Table 1 included in the LCF Žlong dash., corrected optical model Eq. Ž4.9. with m s 1,2,3,4 poles in Table 1 included in the LCF Ždot-dash.. Žb. Same as Ža. but for the Ž3,2,0. § Ž0,0,0. transition at E s 0.4162 eV. Žc. Same as Ža. but for the Ž3,3,0. § Ž0,0,0. transition at E s 0.3872 eV. Žd. Same as Ža. but for the Ž3,3,0. § Ž0,0,0. transition at E s 0.4162 eV.
In this Letter, we shall analyse two transitions Ždetails of the calculation can be found in Ref. w4x., for which the angular distributions show considerable amount of forward scattering ŽFig. 2. ÕX s 3, jX s 2, M X s 0 § Õ s 0, j s 0, M s 0 ,
Ž 2.5a .
at two higher total energies corresponding to the collision energies in the Neumark et al. experiment w1x, E1 s 0.3872 eV and E2 s 0.4162 eV. The four differential cross-sections in Fig. 2 require up to 26 partial waves to converge, N s 26, and in Eq. Ž2.2. we have d - 13. In each case, we have used a quadratic approximation for the phase in Eq. Ž2.1.,
ÕX s 3, jX s 3, M X s 0 § Õ s 0, j s 0, M s 0
Ž 2.5b .
f Ž l . s a l 2 q bl q c
Ž 2.6 .
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
229
Table 1 Values of l m and rm for the four principal Regge poles for the Ž3,2,0. § Ž0,0,0. and Ž3,3,0. § Ž0,0,0. transitions at E s 0.3872 and 0.4162 eV. Also given are parameters a, b and c in the quadratic approximation for the phase f Ž l. in Eq. Ž2.6., lifeangles DQ ' Ž1r2.ImŽ l m . and the values of g ' awImŽ l m .x2 m E s 0.3872 ev 1 2 3 4 E s 0.4162 ev 1 2 3 4 E s 0.3872 ev 1 2 3 4 E s 0.4162 ev 1 2 3 4
Re l m Õ s 0, j s 0
Im l m X X Õ s 3, j s 2
Re rm a s y7.39 Žy2.
Im rm b s 2.35 Žy2.
g c s y3.37 Žy1.
DQ Ždeg.
16.908 15.317 12.062 10.267
0.889 2.633 2.874 3.115
y3.59 Žy2. y1.13 Žy1. y3.54 Žy1. 4.07 Žy1.
4.90 Žy2. 1.38 Žy1. 3.20 Žy1. y7.52 Žy2.
5.84 Žy2. 5.12 Žy1. 6.11 Žy1. 7.17 Žy1.
32.2 10.9 10.0 9.2
Õ s 0, j s 0
Õ s 3, j s 2
a s y7.07 Žy2.
b s 1.28 Žy1.
c s 1.21
19.136 17.167 14.433 12.091
0.886 2.700 2.403 2.464
y4.56 Žy2. 5.75 Žy2. y3.54 Žy1. 1.81 Žy3.
3.92 Žy2. 1.67 Žy1. y1.34 Žy1. 4.88 Žy2.
5.54 Žy2. 5.15 Žy1. 4.08 Žy1. 4.29 Žy1.
Õ s 0, j s 0
Õ s 3, j s 3
a s y6.37 Žy2.
b s y1.14 Žy1.
c s 1.41
16.907 14.873 12.349 10.646
0.886 3.817 3.368 3.496
3.31 Žy2. 2.34 Žy1. 1.06 y0.93
y6.99 Žy2. 7.97 Žy1. y1.24 0.19
5.00 Žy2. 9.27 Žy1. 7.22 Žy1. 7.78 Žy1.
32.3 7.5 8.5 8.1
Õ s 0, j s 0 19.139 17.223 14.855 13.635
Õ s 3, j s 3 0.887 3.031 3.374 2.944
a s y6.35 Žy2. y5.79 Žy2. 1.53 Žy1. y6.94 Žy1. 3.94 Žy1.
b s y9.35 Žy2. 3.41 Žy3. y1.80 Žy1. 3.96 Žy1. y1.71 Žy1.
c s 9.54 Žy1. 5.00 Žy2. 5.83 Žy1. 7.23 Žy1. 5.50 Žy1.
32.3 9.5 8.5 9.6
X
X
X
X
X
X
and identified four resonance poles l m Ž Nres s 4.. We found it convenient to redefine f Ž l. by adding to it Argw q Ž l.x
f Ž l . ™ f Ž l . q Arg q Ž l . ,
Ž 2.7 .
approximate f Ž l. by a quadratic form Ž2.6. and recalculate the Pade´ approximation for sŽ l. in Eq. Ž2.2.. The resulting pole positions l m , residues rm and coefficients a, b and c are given in Table 1. Fig. 3a,c show the moduli of SnJX § n Žcircles., Pade´ approximants Sn X § n Ž l. Žsolid. and smooth terms q Ž l. Ždashed.. The derivatives of Argw Sn X § n Ž l.x and f Ž l. with respect to l are shown in Fig. 3b,d by solid and dashed lines, respectively. Also shown by a dotdashed line is the derivative of the quadratic approximation to f Ž l., 2 a l q b. Next we evaluate the integral in Eq. Ž1.5..
32.3 10.6 11.9 11.6
sis. Thus, we attempt to evaluate the integral Ž1.5. with a rapidly oscillating term exp iw lw q f Ž l.x4 by the stationary phase method. The stationary phase condition reads
w s yf X Ž lS . s y2 a lS y b
Ž 3.1 .
where prime indicates differentiation with respect to l. Eq. Ž3.1. determines the values of the total angular momentum " lS which classically lead to a precession angle w . For a single stationary point lS , the stationary phase method gives
X f Õopt §Õ
Ž w . f sn X § n Ž lS .
2 pl S
1r2
i f Y Ž lS .
= exp i f Ž lS . q wlS 4. Semiclassical optical model Collision partners in a chemical reaction are usually sufficiently heavy to justify semiclassical analy-
4.
Ž 3.2 .
For the transitions Ž2.5. we have found ŽFig. 3b,d. yp - f X Ž l . - 0
Ž 3.3 .
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
230
Our primitive semiclassical approximation Ž3.4. is similar to the optical model of Herschbach w16,17x in which collision partners are represented by hard spheres of Žadjustable. radius R. The classical hardsphere angular distribution is then modified by the quantum probability for the atom B to be transferred between A and C, < s ÕX § Õ Ž lS .< 2 w16,17x. Our optical approximation for the differential cross-section < opt <2 X X sn opt § n Ž u . ' fn § n Ž u .
Ž 3.5 .
yields a similar result but for the classical deflection function
u Ž J . s y2"y1 aJ y b
Ž 3.6 .
where J˜' " l s " Ž J q 1r2. is the classical angular momentum. Semiclassical optical angular distributions for the transition Ž2.5. are shown in Fig. 2 as function of u R ' p y u by dashed lines. The agreement between the exact and optical differential cross-sections is poor, in particular, for small u R . This is expected because, for sufficiently large lifeangles, resonance peaks in Sn X § n Ž l. are destroyed by rotation and replaced in fn X § n Ž u . by exponentially decaying tails w7,8x. Next we correct the model to account for this effect.
5. Lifetime correction function Fig. 3. Ža. Transition Ž3,2,0. § Ž0,0,0., moduli of: S-matrix element S J s SŽ J q1r2. Žcircles.; Pade´ approximant SŽ l. vs. l Žsolid.; smooth part of the Pade´ approximant SŽ l., q Ž l. Ždashed. vs. l. Žb. Transition Ž3,2,0. § Ž0,0,0., derivatives of: the phase of the S-matrix, dArgw SŽ l.xrd l, Žsolid.; the phase of the smooth part of the S-matrix, dArgwexpw i f Ž l.x q Ž l.xrd l, Ždashed.. Also shown by a dot-dashed line is the derivative of the quadratic fit to the smooth phase, 2 a l q b, vs. l.
so that in Eq. Ž1.4. only the k s 0 Žfirst nearside. term has appreciable magnitude, X f Õopt § ÕŽu .
f Ž ik Õ .
y1
Ž 2 p sin u .
y1r2
f˜opt Ž u . exp Ž yipr4 . Ž 3.4 .
where f˜opt Ž u . is given by Eq. Ž3.2..
Mathematically, the optical model Ž3.2. – Ž3.4. fails because the primitive semiclassical approximation of Section 4 breaks down when the stationary point lS approaches one of the poles l m , m s 1,2 . . . Nres . To obtain a uniform approximation for f˜n X § n Ž w ., we N res 1r2 w Ž .xŽ l y first subtract Ý ms 1 l m r m exp i lw q i f l y1 l m . from the integrand of Eq. Ž1.5.. The resulting integral has no poles at l s l m and can be evaluated by the method of Section 4. In the remaining intew grals containing pole contributions l1r2 m r m exp i lw q i f Ž l.xŽ l y l m .y1 we replace the lower integration limit by y`, and express them in terms of the complementary error functions Žsee Appendix A.. The result can be written as
˜X X f˜n X § n Ž w . s f˜nopt § n Ž w . q d fn § n Ž w . .
Ž 4.1 .
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
231
Fig. 4. The LCF d f˜Ž w . for a single pole l0 with wres s 908, a s y0.08 for different values of g ' awIm l0 x 2 , g s 0.01, 0.1, 0.2, 0.3, 1.0. Also shown by dashed lines is the exponential asymptote in Ž4.9.. The exponential tail is drawn where the error of Ž4.12. is less than 10%.
where
and Nres
d f˜n X § n Ž w . s
Ý d f˜nm§ n Ž w . ,
Ž 4.2 .
X
ms1
and
½
d f˜nmX § n Ž w . s l1r2 m r m ip exp i F Ž l m . =erfc Ž yia .
1r2
1 y
4
Ž l m y lS . 1r2
2p
lS y l m yF Y Ž lS .
=exp i F Ž lS .
4
5
.
Ž 4.3 .
Also,
F Ž l . s a l2 q Ž b q w . l q c ,
Ž 4.4 .
lS Ž w . s y Ž w q b . r2 a ,
Ž 4.5 .
erfc Ž z . '
2 p
1r2
`
Hz exp Ž yt
2
. dt
Ž 4.6 .
is the complementary error function w18x. Eqs. Ž4.1., Ž4.2. and Ž4.3., which are exact to semiclassical accuracy, are our main result for a system which, like F q H 2 , has a quadratic phase Ž2.6.. The first term in Eq. Ž4.1. corresponds to the optical model of Section 4, while the second one contains corrections due to the finite lifetimes Žlifeangles. of the resonances. If the resonances are too short-lived for the complexes to rotate before decaying into products d f˜n X § n Ž w . vanishes and the optical model result is recovered. If some of the lifeangles are large, the full scattering amplitude can X be seen as a result of interference between f˜nopt § n and several correction terms. For each pole term, the lifetime correction function ŽLCF. d f˜nmX § n Ž w . Ž4.3. is the difference between the uniform Žerror function. result and the primitive semiclassical approximation of Section 4.
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
232
Consider its behaviour as a function of Im l m . Let wres ' Ž aRe l m q b . be the direct scattering angle corresponding to l s Re l m . Using well-known properties of the error function w18x, erfc Ž z . f Ž p 1r2 z .
y1
exp Ž yz 2 . ,
< z < ™ `,
6. Analysis of F H H 2 angular scattering For the two transitions Ž2.5. and two energies Ž0.3282 and 0.4162 eV., we find that f˜n X § n Ž w . is well extended into the first farside region 1808 - w - 3608 and has almost no structure ŽFig. 5.. The X optical model result f˜nopt § n , shown in Fig. 5 by a dot-dashed line, is contained in the Žfirst nearside.
Ž 4.7 .
we find that for w < wres
d f˜ÕmX § Õ Ž w . f 0 .
Ž 4.8 .
For w 4 wres the LCF describes rotation and decay of the complex, and has a simple exponential form
d f˜ÕmX § Õ Ž w . f 2 p i rm exp iF Ž l m . 4 f const = exp i w Ž Re l m q iIm l m . 4 . Ž 4.9 . Around w f wres we have a transitional region D w where direct scattering cannot be distinguished from immediate decay of a newly formed complex. A long-lived state is formed if l lies in the range D l f Im l0 around Re l m . The width of the corresponding direct angular range gives the size of the transitional region, D w f F Y ŽRe l m ., D l f aŽIm l m .. As Im l m ™ 0, we find D w f a f " Žnote that as " ™ 0 we must have a f " for the classical deflection function Ž3.6. to have a finite slope. and the LCF sharply peaked in the transitional region. As the pole moves away from the real axis, its exponential tail Ž4.9. becomes shorter, while D l grows. At the same time the magnitude of the LCF decreases as Ž . X f˜n X § n Ž w . tends to f˜nopt § n w . The condition for a resonance to produce a distinct exponential contribution to f˜n X § n Ž w . requires 1rIm l m 4 D w , or, equivalently, 2
g ' aw Im l m x < 1 . The LCF for a Žfictitious. pole l0 with wres s 908, r 0 s 1, a s y0.08 and b s c s 0 is shown in Fig. 4 for g s 0.01, 0.1, 0.2, 0.3, 1.0. Note that a well-defined exponential tail disappears already at g f 0.3. Next we apply the method to the title reaction.
Fig. 5. Ža. Transition Ž3,2,0. § Ž0,0,0. at Es 0.3872 eV: the optical model result f˜opt Ž w . Ždot-dashed., the LCF d f˜Ž w . which includes the four poles in Table 1 Žlong dashed. and the full semiclassical amplitude f˜Ž w . s f˜opt Ž w .q d f˜w . Also shown are the resonance angles wres for the four poles Žspikes. and the exponential tail Ž4.9. Ždashed.. Žb. Same as Ža. but for Es 0.4162 eV. Žc. Same as Ža. but for the Ž3,2,0. § Ž0,0,0. transition. Žd. Same as Žc. but for Es 0.4162 eV.
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
region 08 - w - 1808. In all cases, the four-pole LCF shown in Fig. 5 by a dashed line has a transitional region at 608 - w - 1808 and for w - 608 all scattering is direct. In the first farside region, d f˜n X § n Ž w . has an exponential tail which can be attributed to the decay of a single Regge state, l1 f 16.91 q i0.89 at E s 0.3872 eV and l1 f 19.14 q i0.89 at E s 0.4162 eV ŽTable 1.. The forward scattering and nearside–farside oscillations observed in Ref. w4x ŽFig. 2, solid., should therefore be interpreted as a resonance effect. Our semiclassical model ŽFig. 2, dot-dashed. reproduces these oscillations to almost graphical accuracy. Finally, note that although only the leading Ž m s 1. resonance has a considerable angular life Žg f 0.06. we need, in general, all four poles to achieve a good agreement between exact and semiclassical crosssections. Semiclassical results with only the leading pole included in the LCF are shown in Fig. 2 by a long-dashed line.
7. Summary and conclusions We have developed a simple method to estimate resonance effects in reactive Žalso in elastic and inelastic. atom–diatom angular distributions. Introduction of the amplitude f˜n X § n Ž w . allows us ‘unfold’ the scattering amplitude fn X § n Ž u . by separating different rotations of the Jacobi vector R BC ™ A around the total angular momentum. Semiclassically, f˜n X § n Ž w . is decomposed into the direct and resonance contributions Žexcept in the transitional region, where such decomposition is not possible.. A structure in the angular distribution can then be explained as a result of interference between various direct, resonance, nearside and farside contributions. The Pade´ method has been used to accurately obtain the positions and residues of the resonance poles. For the two F q H 2 ™ FH q H transitions Ž2.5. we attribute the small angle scattering as well as the nearside–farside oscillations to the decay of the Regge state corresponding to the leading pole. This agrees with the finding of Dobbyn et al. w5x. Note, however, that a direct comparison with the conclusions of Ref. w4x requires analysis of cross-sections summed over all helicity and rotational quantum numbers. We intend to do this in future work.
233
Acknowledgements One of us ŽD.S.. is grateful to Professor D. Bessis for introduction to the Pade´ method, to Dr. A.J. Dobbyn for useful discussions and to Professor J.N.L. Connor for helpful discussions and hospitality during a visit to Manchester.
Appendix A Consider an integral of the form `
I'
Hy`
exp Ž A l2 q Bl .
l y l0
dl ,
Im l0 ) 0 .
Ž A.1 .
For Im l0 ) 0 we can rewrite ŽA.1. as `
`
Hy` d lH0
I'y
d t exp A l2 q Bl q t Ž l y l0 . .
Ž A.2 . Evaluation of the Gaussian integral over l gives I s i Ž prA .
1r2
`
H0
2
d t exp y Ž B q t . r4 A y t l0 .
Ž A.3 . Finally, a change of variables t ™ Ž t q B .rŽ2'A . q A l 0 yields
(
I s ip exp A l20 q Bl0 erfc Br Ž 2'A . q A l0
ž
(
/
where erfcŽ z . is the complementary error function w18x in Eq. Ž4.7.. Choosing A s a, B s Ž b " u q 2 p n. and l0 s l m , we obtain the first term in Eq. Ž4.3..
References w1x D.M. Neumark, A.M. Wodtke, G.N. Robinson, C.C. Hayden, Y.T. Lee, J. Chem. Phys. 82 Ž1985. 3045. w2x F.J. Aoiz, L. Bannares, V.J. Herrero, V. Sa’ez Ra’banos, K. ˜ Stark, H.-J. Werner, J. Chem. Phys. 102 Ž1995. 9248. w3x J.F. Castillo, D.E. Manolopoulos, K. Stark, H.-J. Werner, J. Chem. Phys. 104 Ž1996. 6531. w4x J.F. Castillo, B. Harthe, H.-J. Werner, F.J. Aoiz, L. Banares, ˜ B. Martinez-Haya, J. Chem. Phys. 109 Ž1998. 7224. w5x A.J. Dobbyn, P. McCabe, J.N.L. Connor, J.F. Castillo, Phys. Chem. Chem. Phys. 1 Ž1999. 1115.
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w6x J.N.L. Connor, P. McCabe, D. Sokolovski, G.C. Schatz, Chem. Phys. Lett. 206 Ž1993. 119. w7x D. Sokolovski, J.N.L. Connor, G.C. Schatz, Chem. Phys. Lett. 238 Ž1995. 127. w8x D. Sokolovski, J.N.L. Connor, G.C. Schatz, J. Chem. Phys. 103 Ž1995. 5979. w9x W.H. Miller, J. Chem. Phys. 53 Ž1970. 1949. w10x G.C. Schatz, A. Kupperman, J. Chem. Phys. 65 Ž1976. 4642. w11x D.M. Brink, G.R. Satcher, Angular Momentum, Clarendon Press, Oxford, 1962. w12x D.M. Brink, Semiclassical Methods for Nucleous–Nucleous Scattering, Cambridge University, Cambridge, 1992.
w13x D. Sokolovski, J.N.L. Connor, Chem. Phys. Lett. 305 Ž1999. 238. w14x G.A. Baker, Jr., The Essentials of Pade´ Approximations, Academic, New York, 1975. w15x D. Bessis, A. Haffad, A.Z. Msezane, Phys. Rev. A 49 Ž1994. 3366. w16x D.R. Herschbach, Appl. Opt. 2 Ž1965. 128, Suppl. w17x D.R. Herschbach, Adv. Chem. Phys. 10 Ž1966. 319. w18x W. Gautschi, in: M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, National Buro of Standards, Applied Mathematics Series, 1964.
Chemical Physics Letters 313 Ž1999. 225–234 www.elsevier.nlrlocatercplett
Semiclassical angular scattering in the F q H 2 ™ HF q H reaction: Regge pole analysis using the Pade´ approximation D. Sokolovski a
a,)
, J.F. Castillo b, C. Tully
a
Theoretical and Computational Physics Research DiÕision, Department of Applied Mathematics and Theoretical Physics, Queen’s UniÕersity of Belfast, Belfast BT7 1NN, UK b Departamento de Quimica Fisica I, UniÕersidad Complutense de Madrid, Madrid 28040, Spain Received 22 March 1999; in final form 24 May 1999
Abstract We apply the complex angular momentum analysis to study state-selective forward scattering in the F q H 2 reaction. The scattering matrix calculated by Castillo et al. is used. The S-matrix element is analytically continued into the complex plane of the total angular momentum with the help of the Pade´ approximation. Resonance ŽRegge. poles are identified and their residues evaluated. The semiclassical optical model is corrected to account for finite lifeangles of leading resonances. For chosen transitions, the forward scattering enhancement is shown to be a resonance effect due to the decay of a Regge state with angular life of about 328. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Reactive angular distributions in the F q H 2 ™ FH q H reaction have been a subject of much attention for more than a decade w1–5x. Their main feature, the state-selective forward angle scattering, was observed by Neumark and co-workers in 1985 w1x. Neumark et al. suggested that the enhanced forward scattering for HF Ž ÕX s 3. was due to the existence of a long-lived state formed during the reaction w1x. However, quasiclassical trajectory studies on the Stark–Werner ŽSW. potential energy surface ŽPES. w2x have demonstrated that classical trajectories producing HF Ž ÕX s 3. also have a tendency to be scattered in the forward direction, thus suggesting a different explanation for the effect. Quantum-mecha) Corresponding author. Fax: q44-1232-239-182; e-mail: [email protected]
nical calculations on the SW PES w3,4x showed even greater forward scattering. Whether enhanced forward scattering in F q H 2 has a direct or resonance origin remained an open question. The authors of Ref. w3x analysed time delays associated with the peak structure in the cumulative reaction probability at zero total angular momentum J versus energy E and concluded that no long-lived state was involved. An additional observation was made by Dobbyn et al. w5x. They have pointed out that, for a light system, the angular life of a resonance can be considerable even if the complex is relatively short-lived. It was also demonstrated that the direct scattering optical model of Herschbach w6x fails to reproduce correctly the small angle scattering for F q H 2 ™ FH q H. Using a more sophisticated nearside–farside analysis w6x the authors of Ref. w5x suggested that the oscillatory structures in the angular distributions were consistent with the presence of
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 0 1 6 - 7
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
226
a single Regge state with a lifeangle of 308–408 at a total energy E s 0.4309 eV. A more rigorous analysis of the problem is required w5x. In this Letter, we present a detailed analysis of resonance contributions to the zero helicity state-tostate differential cross-sections obtained in quantum calculations w4x. In a slightly modified form, we shall apply the semiclassical theory of resonance angular scattering developed in Refs. w7,8x. This Letter is organised as follows: in Section 2, we apply the Poisson sum formula to obtain an integral representation for the reactive scattering amplitude. In Section 3, we use the Pade´ approximation to analytically continue the S-matrix elements into the complex plane of the total angular momentum and identify the resonance poles. In Section 4, we use the Pade´ approximation to construct a semiclassical optical model similar to that of Herschbach w6x. This model will fail if resonances with large lifeangles are present. In Section 5, we define the lifetime correction function ŽLCF. to correct the optical model. In Section 6, we give the results for the F q H 2 ™ FH q H reactions. Section 7 contains our conclusions. 2. Reactive scattering amplitude For an atom–diatom collision, A q BC ™ AB q C the reactive scattering amplitude fn X § n is given by the partial wave series w9,10x, fn X § n Ž u .
Fig. 1. Initial and final orientations of the inelastic Jacobi vector R BC ™ A and the final direction of the reactive Jacobi vector RAB ™ C . The total angular momentum J is perpendicular to the plane of the drawing. In the course of collision R BC ™ A sweeps the angle w Žshaded.. Another final direction of RAB ™ C , leading to the same scattering angle u Ž w sp q u . is shown by a dashed line. Also shown is the reactive scattering angle u R .
Using the semiclassical asymptote of PJ Žcos u . for large J w12x PJ Ž cos u . f
`
s Ž ikn .
y1
Ý Ž J q 1r2.
d MJ X M
Žu .
SnJX § n
,
JsJmin
Ž 1.1 . where n , Ž n X . is a collective index, n ' Ž Õ, j, M ., comprising the vibrational and rotational quantum numbers of the molecule, Õ and j, and the momentum Ž p-. helicity M Žprojection of the molecule’s angular momentum onto the atom–molecule relative velocity.. In addition, J is the total angular momentum quantum number, u is the centre of mass scattering angle ŽFig. 1., kn is the translational wavevector in the incident channel and Jmin ' maxŽ< M <, < M X <.. If both helicities are zero, M s M X s 0, the Wigner matrices d MJ X M Ž u . w11x reduce to the Legendre polynomials PJ , J d 00 Ž u . s PJ Ž cos u . . Ž 1.2 .
1r2
2
Ž J q 1r2. p sin u = cos Ž J q 1r2 . u y pr4 ,
Ž 1.3 .
1rJ - u - p y 1rJ . and applying the Poisson sum formula w12x to Eqs. Ž1.1. and Ž1.2. we can rewrite the scattering amplitude fn X § n Ž u . in an equivalent form fn X § n Ž u . f Ž ik i .
y1
Ž 2 p sin u .
y1r2
`
Ý f˜n § n Ž w k . exp Ž yipr4 y ik pr2.
=
X
ksy`
Ž 1.4 . where f˜n X § n Ž w . '
`
H0 exp Ž i lw . S
X
Õ §Õ
Ž l . l1r2 d l ,
y `- w-` , Sn X § n Ž J q 1r2 . s SnJX § n ,
Ž 1.5 .
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
and k
½
w k Ž u . s Ž y1 . u q p k q 1 y Ž y1 . p k - w k - p Ž k q 1. .
k
5
r2 ,
Ž 1.6 .
We shall use representation Ž1.4. – Ž1.6. to study resonance effects in reactive angular distributions. Semiclassically, the variable w in Eq. Ž1.5. coincides w13x with the angle by which the Jacobi vector R BC ™ A Ži.e. the vector connecting the centre of mass of BC with A. rotates during the collision around the fixed total angular momentum J ŽFig. 1. and f˜n X § n Ž w . is proportional to the probability amplitude for R BC ™ A to rotate by w . How far R BC ™ A will precess around J depends on the reaction mechanism. If the forces between collision partners are mostly repulsive, direct reactive classical trajectories have 0 - w - p and f˜n X § n Ž w . is likely to be contained between w s 0 and w s p. If, on the other hand, the atoms form a long-lived intermediate state, rotation of the decaying complex Žrequired to maintain the total angular momentum J . may carry R BC ™ A to larger w s, and f˜n X § n Ž w . would extend beyond the w0,p x region. Since the full scattering amplitude Ž1.4. is obtained by adding, with appropriate phases, contributions from all rotation angles consistent with the scattering angle u Žthese are given by Eq. Ž1.6.. we expect resonances with sufficiently large lifeangles lead to observable interference effects in reactive angular distributions w7,8x. Next we shall use Eqs. Ž1.4., Ž1.5. and Ž1.6. to investigate the origin of forward scattering of the zero-helicity Ž M s M X s 0. state-to-state angular distributions of the F q H 2 ™ FH q H reaction observed in calculations w3x.
3. Pade´ approximation for the S-matrix. Resonance Regge poles Usually, a compute code evaluates the S-matrix elements SnJX § n , for integer values of J, J s 0,1,2,3 . . . N. To reconstruct from this discrete set of values an analytic function Sn X § n X Ž l. in Eq. Ž1.5., we shall use the Pade´ approximation w14,15x. We note first that, for a heavy system, the S-matrix element is likely to have a phase which rapidly
227
varies with l. To avoid reproducing these rapid oscillations we write Sn X § n Ž l . s exp i f Ž l . sn X § n Ž l . Ž 2.1 . Ž . where f l is a known analytical function chosen so that remaining sn X § n Ž l. varies with l sufficiently slowly. For sn X § n X Ž l. we have a Pade´ approximation d ri Pdy1 Ž l . sn X § n Ž l . s Ý ' Ž 2.2 . Q d Ž l. is1 l y l i where Pdy 1Ž l. and Q d Ž l. are polynomials of l of order d y 1 and d, respectively. A detailed description of the Pade´ method was given by Bessis et al. in Ref. w15x and, to avoid repetition, we shall only review the procedure briefly. Choosing 2 d half-integer values lŽ k ., k s 1,2, . . . ,2 d ( N and using Eqs. Ž2.1. and Ž2.2. yields 2 d equations for 2 d unknowns r i and l i , i s 1,d d ri S ÕX § Õ Ž lŽ k . . exp yi f Ž lŽ k . . s Ý Ž k . , is1 l y l i k s 1,2, . . . 2 d . Ž 2.3 . The non-linear Eq. Ž2.2. can be solved by expressing the function in the r.h.s as a continued fraction w14x. This allows for determination of the coefficients of the polynomials Pdy 1Ž l. and Q d Ž l. in Eq. Ž2.2. and, finally, of the quantities r i and l i . Note that a successful Pade´ approximant should not have any sharp features between the grid points lŽ k .. Also, all physical resonance poles should lie in the first quadrant of the complex l-plane. Under these conditions Eqs. Ž2.1. and Ž2.2. provide an analytic continuation of the S-matrix element in the vicinity of the real l-axis. Poles of the S-matrix close to the real axis correspond to metastable states of the triatomic w7,8x. Separating in Eq. Ž2.3. those, say, Nres Ž Nres - d . poles which correspond to the resonances from those which contribute to the smooth ‘background’ q Ž l., we have Sn X § n Ž l . s exp i f Ž l .
Nres
rm
ms1
l y lm
½Ý
q q Ž l.
5
Ž 2.4 . where q Ž l. includes all remaining terms in Eq. Ž2.3.. The resonance poles in Eq. Ž2.4. may be chosen to ensure that the remaining q Ž l. has as little structure as possible.
228
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
Fig. 2. Ža. Differential cross-sections for the Ž3,2,0. § Ž0,0,0. transition at E s 0.3872 eV vs. u R s p y u : exact PWS Eq. Ž1.1. Žsolid.; optical model Eq. Ž3.5. Žshort dash.; corrected optical model Eq. Ž4.9. with only m s 1 pole in Table 1 included in the LCF Žlong dash., corrected optical model Eq. Ž4.9. with m s 1,2,3,4 poles in Table 1 included in the LCF Ždot-dash.. Žb. Same as Ža. but for the Ž3,2,0. § Ž0,0,0. transition at E s 0.4162 eV. Žc. Same as Ža. but for the Ž3,3,0. § Ž0,0,0. transition at E s 0.3872 eV. Žd. Same as Ža. but for the Ž3,3,0. § Ž0,0,0. transition at E s 0.4162 eV.
In this Letter, we shall analyse two transitions Ždetails of the calculation can be found in Ref. w4x., for which the angular distributions show considerable amount of forward scattering ŽFig. 2. ÕX s 3, jX s 2, M X s 0 § Õ s 0, j s 0, M s 0 ,
Ž 2.5a .
at two higher total energies corresponding to the collision energies in the Neumark et al. experiment w1x, E1 s 0.3872 eV and E2 s 0.4162 eV. The four differential cross-sections in Fig. 2 require up to 26 partial waves to converge, N s 26, and in Eq. Ž2.2. we have d - 13. In each case, we have used a quadratic approximation for the phase in Eq. Ž2.1.,
ÕX s 3, jX s 3, M X s 0 § Õ s 0, j s 0, M s 0
Ž 2.5b .
f Ž l . s a l 2 q bl q c
Ž 2.6 .
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
229
Table 1 Values of l m and rm for the four principal Regge poles for the Ž3,2,0. § Ž0,0,0. and Ž3,3,0. § Ž0,0,0. transitions at E s 0.3872 and 0.4162 eV. Also given are parameters a, b and c in the quadratic approximation for the phase f Ž l. in Eq. Ž2.6., lifeangles DQ ' Ž1r2.ImŽ l m . and the values of g ' awImŽ l m .x2 m E s 0.3872 ev 1 2 3 4 E s 0.4162 ev 1 2 3 4 E s 0.3872 ev 1 2 3 4 E s 0.4162 ev 1 2 3 4
Re l m Õ s 0, j s 0
Im l m X X Õ s 3, j s 2
Re rm a s y7.39 Žy2.
Im rm b s 2.35 Žy2.
g c s y3.37 Žy1.
DQ Ždeg.
16.908 15.317 12.062 10.267
0.889 2.633 2.874 3.115
y3.59 Žy2. y1.13 Žy1. y3.54 Žy1. 4.07 Žy1.
4.90 Žy2. 1.38 Žy1. 3.20 Žy1. y7.52 Žy2.
5.84 Žy2. 5.12 Žy1. 6.11 Žy1. 7.17 Žy1.
32.2 10.9 10.0 9.2
Õ s 0, j s 0
Õ s 3, j s 2
a s y7.07 Žy2.
b s 1.28 Žy1.
c s 1.21
19.136 17.167 14.433 12.091
0.886 2.700 2.403 2.464
y4.56 Žy2. 5.75 Žy2. y3.54 Žy1. 1.81 Žy3.
3.92 Žy2. 1.67 Žy1. y1.34 Žy1. 4.88 Žy2.
5.54 Žy2. 5.15 Žy1. 4.08 Žy1. 4.29 Žy1.
Õ s 0, j s 0
Õ s 3, j s 3
a s y6.37 Žy2.
b s y1.14 Žy1.
c s 1.41
16.907 14.873 12.349 10.646
0.886 3.817 3.368 3.496
3.31 Žy2. 2.34 Žy1. 1.06 y0.93
y6.99 Žy2. 7.97 Žy1. y1.24 0.19
5.00 Žy2. 9.27 Žy1. 7.22 Žy1. 7.78 Žy1.
32.3 7.5 8.5 8.1
Õ s 0, j s 0 19.139 17.223 14.855 13.635
Õ s 3, j s 3 0.887 3.031 3.374 2.944
a s y6.35 Žy2. y5.79 Žy2. 1.53 Žy1. y6.94 Žy1. 3.94 Žy1.
b s y9.35 Žy2. 3.41 Žy3. y1.80 Žy1. 3.96 Žy1. y1.71 Žy1.
c s 9.54 Žy1. 5.00 Žy2. 5.83 Žy1. 7.23 Žy1. 5.50 Žy1.
32.3 9.5 8.5 9.6
X
X
X
X
X
X
and identified four resonance poles l m Ž Nres s 4.. We found it convenient to redefine f Ž l. by adding to it Argw q Ž l.x
f Ž l . ™ f Ž l . q Arg q Ž l . ,
Ž 2.7 .
approximate f Ž l. by a quadratic form Ž2.6. and recalculate the Pade´ approximation for sŽ l. in Eq. Ž2.2.. The resulting pole positions l m , residues rm and coefficients a, b and c are given in Table 1. Fig. 3a,c show the moduli of SnJX § n Žcircles., Pade´ approximants Sn X § n Ž l. Žsolid. and smooth terms q Ž l. Ždashed.. The derivatives of Argw Sn X § n Ž l.x and f Ž l. with respect to l are shown in Fig. 3b,d by solid and dashed lines, respectively. Also shown by a dotdashed line is the derivative of the quadratic approximation to f Ž l., 2 a l q b. Next we evaluate the integral in Eq. Ž1.5..
32.3 10.6 11.9 11.6
sis. Thus, we attempt to evaluate the integral Ž1.5. with a rapidly oscillating term exp iw lw q f Ž l.x4 by the stationary phase method. The stationary phase condition reads
w s yf X Ž lS . s y2 a lS y b
Ž 3.1 .
where prime indicates differentiation with respect to l. Eq. Ž3.1. determines the values of the total angular momentum " lS which classically lead to a precession angle w . For a single stationary point lS , the stationary phase method gives
X f Õopt §Õ
Ž w . f sn X § n Ž lS .
2 pl S
1r2
i f Y Ž lS .
= exp i f Ž lS . q wlS 4. Semiclassical optical model Collision partners in a chemical reaction are usually sufficiently heavy to justify semiclassical analy-
4.
Ž 3.2 .
For the transitions Ž2.5. we have found ŽFig. 3b,d. yp - f X Ž l . - 0
Ž 3.3 .
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
230
Our primitive semiclassical approximation Ž3.4. is similar to the optical model of Herschbach w16,17x in which collision partners are represented by hard spheres of Žadjustable. radius R. The classical hardsphere angular distribution is then modified by the quantum probability for the atom B to be transferred between A and C, < s ÕX § Õ Ž lS .< 2 w16,17x. Our optical approximation for the differential cross-section < opt <2 X X sn opt § n Ž u . ' fn § n Ž u .
Ž 3.5 .
yields a similar result but for the classical deflection function
u Ž J . s y2"y1 aJ y b
Ž 3.6 .
where J˜' " l s " Ž J q 1r2. is the classical angular momentum. Semiclassical optical angular distributions for the transition Ž2.5. are shown in Fig. 2 as function of u R ' p y u by dashed lines. The agreement between the exact and optical differential cross-sections is poor, in particular, for small u R . This is expected because, for sufficiently large lifeangles, resonance peaks in Sn X § n Ž l. are destroyed by rotation and replaced in fn X § n Ž u . by exponentially decaying tails w7,8x. Next we correct the model to account for this effect.
5. Lifetime correction function Fig. 3. Ža. Transition Ž3,2,0. § Ž0,0,0., moduli of: S-matrix element S J s SŽ J q1r2. Žcircles.; Pade´ approximant SŽ l. vs. l Žsolid.; smooth part of the Pade´ approximant SŽ l., q Ž l. Ždashed. vs. l. Žb. Transition Ž3,2,0. § Ž0,0,0., derivatives of: the phase of the S-matrix, dArgw SŽ l.xrd l, Žsolid.; the phase of the smooth part of the S-matrix, dArgwexpw i f Ž l.x q Ž l.xrd l, Ždashed.. Also shown by a dot-dashed line is the derivative of the quadratic fit to the smooth phase, 2 a l q b, vs. l.
so that in Eq. Ž1.4. only the k s 0 Žfirst nearside. term has appreciable magnitude, X f Õopt § ÕŽu .
f Ž ik Õ .
y1
Ž 2 p sin u .
y1r2
f˜opt Ž u . exp Ž yipr4 . Ž 3.4 .
where f˜opt Ž u . is given by Eq. Ž3.2..
Mathematically, the optical model Ž3.2. – Ž3.4. fails because the primitive semiclassical approximation of Section 4 breaks down when the stationary point lS approaches one of the poles l m , m s 1,2 . . . Nres . To obtain a uniform approximation for f˜n X § n Ž w ., we N res 1r2 w Ž .xŽ l y first subtract Ý ms 1 l m r m exp i lw q i f l y1 l m . from the integrand of Eq. Ž1.5.. The resulting integral has no poles at l s l m and can be evaluated by the method of Section 4. In the remaining intew grals containing pole contributions l1r2 m r m exp i lw q i f Ž l.xŽ l y l m .y1 we replace the lower integration limit by y`, and express them in terms of the complementary error functions Žsee Appendix A.. The result can be written as
˜X X f˜n X § n Ž w . s f˜nopt § n Ž w . q d fn § n Ž w . .
Ž 4.1 .
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
231
Fig. 4. The LCF d f˜Ž w . for a single pole l0 with wres s 908, a s y0.08 for different values of g ' awIm l0 x 2 , g s 0.01, 0.1, 0.2, 0.3, 1.0. Also shown by dashed lines is the exponential asymptote in Ž4.9.. The exponential tail is drawn where the error of Ž4.12. is less than 10%.
where
and Nres
d f˜n X § n Ž w . s
Ý d f˜nm§ n Ž w . ,
Ž 4.2 .
X
ms1
and
½
d f˜nmX § n Ž w . s l1r2 m r m ip exp i F Ž l m . =erfc Ž yia .
1r2
1 y
4
Ž l m y lS . 1r2
2p
lS y l m yF Y Ž lS .
=exp i F Ž lS .
4
5
.
Ž 4.3 .
Also,
F Ž l . s a l2 q Ž b q w . l q c ,
Ž 4.4 .
lS Ž w . s y Ž w q b . r2 a ,
Ž 4.5 .
erfc Ž z . '
2 p
1r2
`
Hz exp Ž yt
2
. dt
Ž 4.6 .
is the complementary error function w18x. Eqs. Ž4.1., Ž4.2. and Ž4.3., which are exact to semiclassical accuracy, are our main result for a system which, like F q H 2 , has a quadratic phase Ž2.6.. The first term in Eq. Ž4.1. corresponds to the optical model of Section 4, while the second one contains corrections due to the finite lifetimes Žlifeangles. of the resonances. If the resonances are too short-lived for the complexes to rotate before decaying into products d f˜n X § n Ž w . vanishes and the optical model result is recovered. If some of the lifeangles are large, the full scattering amplitude can X be seen as a result of interference between f˜nopt § n and several correction terms. For each pole term, the lifetime correction function ŽLCF. d f˜nmX § n Ž w . Ž4.3. is the difference between the uniform Žerror function. result and the primitive semiclassical approximation of Section 4.
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
232
Consider its behaviour as a function of Im l m . Let wres ' Ž aRe l m q b . be the direct scattering angle corresponding to l s Re l m . Using well-known properties of the error function w18x, erfc Ž z . f Ž p 1r2 z .
y1
exp Ž yz 2 . ,
< z < ™ `,
6. Analysis of F H H 2 angular scattering For the two transitions Ž2.5. and two energies Ž0.3282 and 0.4162 eV., we find that f˜n X § n Ž w . is well extended into the first farside region 1808 - w - 3608 and has almost no structure ŽFig. 5.. The X optical model result f˜nopt § n , shown in Fig. 5 by a dot-dashed line, is contained in the Žfirst nearside.
Ž 4.7 .
we find that for w < wres
d f˜ÕmX § Õ Ž w . f 0 .
Ž 4.8 .
For w 4 wres the LCF describes rotation and decay of the complex, and has a simple exponential form
d f˜ÕmX § Õ Ž w . f 2 p i rm exp iF Ž l m . 4 f const = exp i w Ž Re l m q iIm l m . 4 . Ž 4.9 . Around w f wres we have a transitional region D w where direct scattering cannot be distinguished from immediate decay of a newly formed complex. A long-lived state is formed if l lies in the range D l f Im l0 around Re l m . The width of the corresponding direct angular range gives the size of the transitional region, D w f F Y ŽRe l m ., D l f aŽIm l m .. As Im l m ™ 0, we find D w f a f " Žnote that as " ™ 0 we must have a f " for the classical deflection function Ž3.6. to have a finite slope. and the LCF sharply peaked in the transitional region. As the pole moves away from the real axis, its exponential tail Ž4.9. becomes shorter, while D l grows. At the same time the magnitude of the LCF decreases as Ž . X f˜n X § n Ž w . tends to f˜nopt § n w . The condition for a resonance to produce a distinct exponential contribution to f˜n X § n Ž w . requires 1rIm l m 4 D w , or, equivalently, 2
g ' aw Im l m x < 1 . The LCF for a Žfictitious. pole l0 with wres s 908, r 0 s 1, a s y0.08 and b s c s 0 is shown in Fig. 4 for g s 0.01, 0.1, 0.2, 0.3, 1.0. Note that a well-defined exponential tail disappears already at g f 0.3. Next we apply the method to the title reaction.
Fig. 5. Ža. Transition Ž3,2,0. § Ž0,0,0. at Es 0.3872 eV: the optical model result f˜opt Ž w . Ždot-dashed., the LCF d f˜Ž w . which includes the four poles in Table 1 Žlong dashed. and the full semiclassical amplitude f˜Ž w . s f˜opt Ž w .q d f˜w . Also shown are the resonance angles wres for the four poles Žspikes. and the exponential tail Ž4.9. Ždashed.. Žb. Same as Ža. but for Es 0.4162 eV. Žc. Same as Ža. but for the Ž3,2,0. § Ž0,0,0. transition. Žd. Same as Žc. but for Es 0.4162 eV.
D. SokoloÕski et al.r Chemical Physics Letters 313 (1999) 225–234
region 08 - w - 1808. In all cases, the four-pole LCF shown in Fig. 5 by a dashed line has a transitional region at 608 - w - 1808 and for w - 608 all scattering is direct. In the first farside region, d f˜n X § n Ž w . has an exponential tail which can be attributed to the decay of a single Regge state, l1 f 16.91 q i0.89 at E s 0.3872 eV and l1 f 19.14 q i0.89 at E s 0.4162 eV ŽTable 1.. The forward scattering and nearside–farside oscillations observed in Ref. w4x ŽFig. 2, solid., should therefore be interpreted as a resonance effect. Our semiclassical model ŽFig. 2, dot-dashed. reproduces these oscillations to almost graphical accuracy. Finally, note that although only the leading Ž m s 1. resonance has a considerable angular life Žg f 0.06. we need, in general, all four poles to achieve a good agreement between exact and semiclassical crosssections. Semiclassical results with only the leading pole included in the LCF are shown in Fig. 2 by a long-dashed line.
7. Summary and conclusions We have developed a simple method to estimate resonance effects in reactive Žalso in elastic and inelastic. atom–diatom angular distributions. Introduction of the amplitude f˜n X § n Ž w . allows us ‘unfold’ the scattering amplitude fn X § n Ž u . by separating different rotations of the Jacobi vector R BC ™ A around the total angular momentum. Semiclassically, f˜n X § n Ž w . is decomposed into the direct and resonance contributions Žexcept in the transitional region, where such decomposition is not possible.. A structure in the angular distribution can then be explained as a result of interference between various direct, resonance, nearside and farside contributions. The Pade´ method has been used to accurately obtain the positions and residues of the resonance poles. For the two F q H 2 ™ FH q H transitions Ž2.5. we attribute the small angle scattering as well as the nearside–farside oscillations to the decay of the Regge state corresponding to the leading pole. This agrees with the finding of Dobbyn et al. w5x. Note, however, that a direct comparison with the conclusions of Ref. w4x requires analysis of cross-sections summed over all helicity and rotational quantum numbers. We intend to do this in future work.
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Acknowledgements One of us ŽD.S.. is grateful to Professor D. Bessis for introduction to the Pade´ method, to Dr. A.J. Dobbyn for useful discussions and to Professor J.N.L. Connor for helpful discussions and hospitality during a visit to Manchester.
Appendix A Consider an integral of the form `
I'
Hy`
exp Ž A l2 q Bl .
l y l0
dl ,
Im l0 ) 0 .
Ž A.1 .
For Im l0 ) 0 we can rewrite ŽA.1. as `
`
Hy` d lH0
I'y
d t exp A l2 q Bl q t Ž l y l0 . .
Ž A.2 . Evaluation of the Gaussian integral over l gives I s i Ž prA .
1r2
`
H0
2
d t exp y Ž B q t . r4 A y t l0 .
Ž A.3 . Finally, a change of variables t ™ Ž t q B .rŽ2'A . q A l 0 yields
(
I s ip exp A l20 q Bl0 erfc Br Ž 2'A . q A l0
ž
(
/
where erfcŽ z . is the complementary error function w18x in Eq. Ž4.7.. Choosing A s a, B s Ž b " u q 2 p n. and l0 s l m , we obtain the first term in Eq. Ž4.3..
References w1x D.M. Neumark, A.M. Wodtke, G.N. Robinson, C.C. Hayden, Y.T. Lee, J. Chem. Phys. 82 Ž1985. 3045. w2x F.J. Aoiz, L. Bannares, V.J. Herrero, V. Sa’ez Ra’banos, K. ˜ Stark, H.-J. Werner, J. Chem. Phys. 102 Ž1995. 9248. w3x J.F. Castillo, D.E. Manolopoulos, K. Stark, H.-J. Werner, J. Chem. Phys. 104 Ž1996. 6531. w4x J.F. Castillo, B. Harthe, H.-J. Werner, F.J. Aoiz, L. Banares, ˜ B. Martinez-Haya, J. Chem. Phys. 109 Ž1998. 7224. w5x A.J. Dobbyn, P. McCabe, J.N.L. Connor, J.F. Castillo, Phys. Chem. Chem. Phys. 1 Ž1999. 1115.
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w6x J.N.L. Connor, P. McCabe, D. Sokolovski, G.C. Schatz, Chem. Phys. Lett. 206 Ž1993. 119. w7x D. Sokolovski, J.N.L. Connor, G.C. Schatz, Chem. Phys. Lett. 238 Ž1995. 127. w8x D. Sokolovski, J.N.L. Connor, G.C. Schatz, J. Chem. Phys. 103 Ž1995. 5979. w9x W.H. Miller, J. Chem. Phys. 53 Ž1970. 1949. w10x G.C. Schatz, A. Kupperman, J. Chem. Phys. 65 Ž1976. 4642. w11x D.M. Brink, G.R. Satcher, Angular Momentum, Clarendon Press, Oxford, 1962. w12x D.M. Brink, Semiclassical Methods for Nucleous–Nucleous Scattering, Cambridge University, Cambridge, 1992.
w13x D. Sokolovski, J.N.L. Connor, Chem. Phys. Lett. 305 Ž1999. 238. w14x G.A. Baker, Jr., The Essentials of Pade´ Approximations, Academic, New York, 1975. w15x D. Bessis, A. Haffad, A.Z. Msezane, Phys. Rev. A 49 Ž1994. 3366. w16x D.R. Herschbach, Appl. Opt. 2 Ž1965. 128, Suppl. w17x D.R. Herschbach, Adv. Chem. Phys. 10 Ž1966. 319. w18x W. Gautschi, in: M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, National Buro of Standards, Applied Mathematics Series, 1964.