Parametrized S matrix models of elastic scattering in chemically reactive systems

The Journal of Chemical Physics

2013

Three new contributions to the complex angular momentum (CAM) theory of differential cross sections (DCSs) for chemical reactions are reported. They exploit recent advances in the Padé reconstruction of a scattering (S) matrix in a region surrounding the ReJ axis, where J is the total angular momentum quantum variable, starting from the discrete values, J = 0, 1, 2, .... In particular, use is made of Padé continuations obtained by Sokolovski, Castillo, and Tully [Chem. Phys. Lett. 313, 225 (1999)] for the S matrix of the benchmark F + H2(v(i) = 0, j(i) = 0, m(i) = 0) → FH(v(f) = 3, j(f) = 3, m(f) = 0) + H reaction. Here v(i), j(i), m(i) and v(f), j(f), m(f) are the initial and final vibrational, rotational, and helicity quantum numbers, respectively. The three contributions are: (1) A new exact decomposition of the partial wave (PW) S matrix is introduced, which is called the QP decomposition. The P part contains information on the Regge poles. The Q part is then constructed exactly by subtracting a rapidly oscillating phase and the PW P matrix from the input PW S matrix. After a simple modification, it is found that the corresponding scattering subamplitudes provide insight into the angular-scattering dynamics using simple partial wave series (PWS) computations. It is shown that the leading n = 0 Regge pole contributes to the small-angle scattering in the centre-of-mass frame. (2) The Q matrix part of the QP decomposition has simpler properties than the input S matrix. This fact is exploited to deduce a parametrized (analytic) formula for the PW S matrix in which all terms have a direct physical interpretation. This is a long sort-after goal in reaction dynamics, and in particular for the state-to-state F + H2 reaction. (3) The first definitive test is reported for the accuracy of a uniform semiclassical (asymptotic) CAM theory for a DCS based on the Watson transformation. The parametrized S matrix obtained in contribution (2) is used in both the PW and semiclassical parts of the calculation. Powerful uniform asymptotic approximations are employed for the background integral; they allow for the proximity of a Regge pole and a saddle point. The CAM DCS agrees well with the PWS DCS, across the whole angular range, except close to the forward and backward directions, where, as expected, the CAM theory becomes non-uniform. At small angles, θ(R) ≲ 40°, the PWS DCS can be reproduced using a nearside semiclassical subamplitude, which allows for a pole being close to a saddle point, plus the farside surface wave of the n = 0 pole sub-subamplitude, with the oscillations in the DCS arising from nearside-farside interference. This proves that the n = 0 Regge resonance pole contributes to the small-angle scattering.

Section: H Results For the F + H 2 Reactionmentioning

confidence: 99%

Resonance Regge poles and the state-to-state F + H2 reaction: QP decomposition, parametrized S matrix, and semiclassical complex angular momentum analysis of the angular scattering

The Journal of Chemical Physics

2013

“…Next we examine Figure , which shows plots of the DCSs for the pole, direct, and refc subamplitudes together with the full uniform Watson/CAM DCSs for the three transitions. We note the following: The DCS for the pole subamplitude is small at large angles, where it possesses the characteristic oscillations of a backward glory (not visible in Figure c). − At smaller angles, the oscillations damp out and the pole DCS becomes monotonic and comparable in magnitude to the full CAM DCS. The direct DCS is small at angles close to the forward direction but becomes comparable to the full CAM DCS at sideward and large angles. For the 300 and 310 cases in Figure a,b, the DCSs for the refc subamplitude are significant at small and sideward angles but then become very small at large angles. This is because the residue and erfc subamplitudes cancel in this angular range; the cancellation occurs when | u n (θ R )| ≫ 1 (see the appendix of ref ).…”

Section: Application To the F + H2 Reactionmentioning

confidence: 88%

State-to-State F + H₂ Reaction at E_trans = 0.04088 eV: QP Decomposition, Parametrized S Matrix Incorporating Regge Poles, and Uniform Asymptotic Complex Angular Momentum Analysis of the Angular Scattering

Xiao

2016

J. Phys. Chem. A

We report two new contributions for understanding the quantum dynamics of the benchmark state-to-state reaction, F + H2(vi, ji, mi) → FH(vf, jf, mf) + H, where (vi, ji, mi) and (vf, jf, mf) are the initial and final vibrational, rotational, and helicity quantum numbers, respectively. We analyze product differential cross sections (DCSs) for the transitions, 000 → 300, 000 → 310, and 000 → 320, at a translational energy of 0.04088 eV using the potential energy surface of Fu-Xu-Zhang. The two new contributions are as follows: (1) We exploit the recently introduced QP decomposition of J. N. L. Connor [ J. Chem. Phys . 2013 , 138 , 124310 ] to transform numerical partial-wave scattering (S) matrix elements for the three transitions into parametrized (analytic) formulas, in which all terms in the three parametrized S matrices have a direct physical interpretation. In particular, they contain the positions and residues of Regge poles in the first quadrant of the complex angular momentum (CAM) plane. We obtain very close agreement between the values of the parametrized and numerical S matrix elements. (2) We then apply a uniform asymptotic Watson/CAM theory, which allows a Regge pole to be close to a saddle point. It uses the parametrized S matrices and is applied to the partial wave series (PWS) representation for the scattering amplitude to understand structure in a DCS in terms of three contributing subamplitudes. We prove using this powerful CAM theory that resonance Regge poles contribute to the small-angle scattering in the DCSs for all three transitions, with the oscillations at larger angles arising from nearside-farside interference. We obtain very good agreement between the uniform asymptotic Watson/CAM DCSs and the corresponding PWS DCSs, except for angles close to the forward and backward directions, where (as expected) the Watson/CAM formulas become nonuniform.

“…The four parameterized S matrices are adapted from previous research on elastic scattering in the presence of open inelastic and reactive channels. 4 Each parameterized S matrix depends on four parameters, which have a clear physical interpretation.…”

Section: Introductionmentioning

confidence: 99%

Angular scattering using parameterized Smatrix elements for the H + D₂(v_i= 0, j_i= 0) → HD(v_f= 3, j_f= 0) + D reaction: an example of Heisenberg's Smatrix programme

Xiao

Phys. Chem. Chem. Phys.

2011

A neglected topic in the theory of reactive scattering is the use of parameterized scattering (S) matrix elements to calculate differential cross sections (DCSs). We construct four simple parameterizations, whose moduli are smooth step-functions and whose phases are quadratic functions of the total angular momentum quantum number. Application is made to forward glory scattering in the DCS of the H + D(2)(v(i) = 0, j(i) = 0) → HD(v(f) = 3, j(f) = 0) + D reaction at a translational energy of 1.81 eV, where v and j are vibrational and rotational quantum numbers respectively. The parameterized S matrix elements can reproduce the forward scattering for centre-of-mass reactive scattering angles up to 30° and can identify the total angular momenta (equivalently, impact parameters) that contribute to the glory. The theoretical techniques employed to analyze structure in the DCS include: nearside-farside theory, local angular momentum theory--in both cases incorporating resummations of the partial wave series representation of the scattering amplitude--and the uniform semiclassical theory of forward glory scattering. Our approach is an example of Heisenberg's S matrix programme, in which no potential energy surface is used. Our calculations for the DCS using the four parameterized S matrix elements are counterexamples to the following universal statements often found in the chemical physics literature: "every molecular scattering investigation needs detailed information about the interaction potential," and "an accurate potential energy surface is an essential element in carrying out simulations of a chemical reaction". Both these statements are false.