×

Extracting \(S\)-matrix poles for resonances from numerical scattering data: type-II Padé reconstruction. (English) Zbl 1216.81134

Summary: We present a FORTRAN 77 code for evaluation of resonance pole positions and residues of a numerical scattering matrix element in the complex energy (CE) as well as in the complex angular momentum (CAM) planes. Analytical continuation of the \(S\)-matrix element is performed by constructing a type-II Padé approximant from given physical values [D. Bessis, A. Haffad and A. Z. Msezane, “Momentum-transfer dispersion relations for electron-atom cross sections”, Phys. Rev. A 42, No. 5, 3366–3375 (1994); D. Vrinceanu, A. Z. Msezane, D. Bessis, J. N. L. Connor and D. Sokolovski, “Padé reconstruction of Regge poles from scattering matrix data for chemical reactions”, Chem. Phys. Lett. 324, 311 (2000); D. Sokolovski and A. Z. Msezane, “Semiclassical complex angular momentum theory and Padé reconstruction for resonances, rainbows, and reaction thresholds”, Phys. Rev. A 70, No. 3, Paper No. 032710, 12 p. (2004)]. The algorithm involves iterative ‘preconditioning’ of the numerical data by extracting its rapidly oscillating potential phase component. The code has the capability of adding non-analytical noise to the numerical data in order to select ‘true’ physical poles, investigate their stability and evaluate the accuracy of the reconstruction. It has an option of employing multiple-precision (MPFUN) package [ACM Trans. Math. Softw. 19, No. 3, 288–319 (1993; Zbl 0889.68015)] developed by D. H. Bailey wherever double precision calculations fail due to a large number of input partial waves (energies) involved. The code has been successfully tested on several models, as well as the \(\text F + \text H_{2} \rightarrow \text{HF} + \text{H}, \text{F} + \text{HD} \rightarrow \text{HF} + \text{D}, \text{Cl} + \text{HCl} \rightarrow \text{ClH} + \text{Cl}\) and \(\text{H} + \text D_{2} \rightarrow \text{HD} + \text{D}\) reactions. Some detailed examples are given in the text.

MSC:

81U20 \(S\)-matrix theory, etc. in quantum theory
81V55 Molecular physics
81V45 Atomic physics
35P25 Scattering theory for PDEs
81-04 Software, source code, etc. for problems pertaining to quantum theory
81T80 Simulation and numerical modelling (quantum field theory) (MSC2010)
41A21 Padé approximation

Citations:

Zbl 0889.68015
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Harich, S. A.; Dai, D. X.; Wang, C. C.; Wang, X.; Chao, S. D.; Skodje, R. T., Forward scattering due to slow-down of the intermediate in the H + HD ← D + \(H_2\) reaction, Nature, 419, 281 (2002)
[2] Clary, D. C., Quantum theory of reaction dynamics, Science, 279, 1879 (1998)
[3] Skouteris, D.; Manolopoulos, D. E.; Bian, W.; Werner, H.-J.; Lai, L.-H.; Liu, K., Van der Waals interactions in the Cl + HD reaction, Science, 286, 1713 (1999)
[4] Casavecchia, P.; Balucani, N.; Volpi, G. G., Cross beam studies of reaction dynamics, Annu. Rev. Phys. Chem., 50, 347 (1999)
[5] Miller, W. H., Classical-limit quantum mechanics and the theory of molecular collisions, Adv. Chem. Phys., 25, 69 (1974)
[6] Skouteris, D.; Castillo, J. F.; Manolopoulos, D. E., ABC: a quantum reactive scattering problem, Comput. Phys. Commun., 133, 128 (2000) · Zbl 0970.81091
[7] Althorpe, S. C.; Fernández-Alonso, F.; Bean, B. D.; Ayers, J. D.; Pomerantz, A. E.; Zare, R. N.; Wrede, E., Observation and interpretation of a time-delayed mechanism in hydrogen exchange reaction, Nature, 416, 67 (2002)
[8] Aquilanti, V.; Cavalli, S.; De Fazio, D., Hyperquantization algorithm. I. Theory for triatomic systems, J. Chem. Phys., 109, 3792 (1998)
[9] Aquilanti, V.; Cavalli, S.; De Fazio, D.; Volpi, A.; Aguilar, A.; Giménez, X.; Lucas, J. F., Exact reaction dynamics by the hyperquantization algorithm: integral and differential cross sections for \(F + H_2\), including long-range and spin-orbit effects, Phys. Chem. Chem. Phys., 4, 401 (2002)
[10] Connor, J. N.L., Molecular collisions and the semiclassical approximation, Meldola Medal Lecture, Chem. Soc. Rev., 5, 125 (1976)
[11] Connor, J. N.L.; Jakubetz, W.; Sukumar, C. V., Exact quantum and semiclassical calculation of the positions and residues of Regge poles for interatomic potentials, J. Phys. B, 9, 1783 (1976)
[12] Connor, J. N.L.; Jakubetz, W., Rainbow scattering in atomic collisions: A Regge pole analysis, Mol. Phys., 35, 949 (1978)
[13] Connor, J. N.L.; Mackay, D. C.; Sukumar, C. V., Quantum and semiclassical calculation of Regge pole positions and residues for complex optical potentials, J. Phys. B, 12, L5151 (1979)
[14] Connor, J. N.L., Semiclassical theory of elastic scattering, (Child, M. S., Semiclassical Methods in Molecular Scattering and Spectroscopy, Proceedings of the NATO Advanced Study Institute. Semiclassical Methods in Molecular Scattering and Spectroscopy, Proceedings of the NATO Advanced Study Institute, Cambridge, England, September 1979 (1980), Reidel: Reidel Dordrecht, The Netherlands), 45 · Zbl 0697.65006
[15] Connor, J. N.L.; Farrelly, D.; Mackay, D. C., Complex angular momentum analysis of diffraction scattering in atomic collisions, J. Chem. Phys., 74, 3278 (1981)
[16] Thylwe, K.-E.; Connor, J. N.L., A complex angular momentum theory of modified Coulomb scattering, J. Phys. A, 18, 2957 (1985)
[17] Connor, J. N.L.; Mackay, D. C.; Thylwe, K.-E., Computational study and complex angular momentum analysis of elastic scattering for complex optical potentials, J. Chem. Phys., 85, 6368 (1986)
[18] Connor, J. N.L.; Thylwe, K.-E., Theory of large angle elastic differential cross sections for complex optical potentials: Semiclassical calculations using partial waves, l-windows, saddles and poles, J. Chem. Phys., 86, 188 (1987)
[19] Connor, J. N.L., New theoretical methods for molecular collisions: The complex angular momentum approach, J. Chem. Soc. Faraday Trans., 86, 1627 (1990) · Zbl 0697.65006
[20] McCabe, P.; Connor, J. N.L.; Thylwe, K.-E., Complex angular momentum theory of molecular collisions: New phase rules for rotationally inelastic diffraction scattering in atom homonuclear-molecule collisions, J. Chem. Phys., 98, 2947 (1993)
[21] Brink, D. M., Semi-Classical Methods in Nucleus-Nucleus Scattering (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0586.47004
[22] Sokolovski, D.; Msezane, A. Z., Semiclassical complex angular momentum theory and Padé reconstruction for resonances, rainbows, and reaction thresholds, Phys. Rev. A, 70, 032710 (2004)
[23] Vrinceanu, D.; Msezane, A. Z.; Bessis, D.; Connor, J. N.L.; Sokolovski, D., Padé reconstruction of Regge poles from scattering matrix data for chemical reactions, Chem. Phys. Lett., 324, 311 (2000)
[24] D. Sokolovski, S. Sen, On the type II Padé reconstruction of a scattering matrix element, in: S.K. Sen, D. Sokolovski, J.N.L. Connor (Eds.), Semiclassical and Other Methods for Understanding Molecular Collisions and Chemical Reactions, Collaborative Computational Project on Molecular Quantum Dynamics (CCP6), Daresbury, UK, 2005, p. 104.; D. Sokolovski, S. Sen, On the type II Padé reconstruction of a scattering matrix element, in: S.K. Sen, D. Sokolovski, J.N.L. Connor (Eds.), Semiclassical and Other Methods for Understanding Molecular Collisions and Chemical Reactions, Collaborative Computational Project on Molecular Quantum Dynamics (CCP6), Daresbury, UK, 2005, p. 104.
[25] Sokolovski, D.; Connor, J. N.L.; Schatz, G. C., New uniform semiclassical theory of resonance angular scattering for reactive molecular collisions, Chem. Phys. Lett., 238, 127 (1995)
[26] Sokolovski, D.; Connor, J. N.L.; Schatz, G. C., Complex angular momentum analysis of resonance scattering in the Cl + HCl → ClH + Cl reaction, J. Chem. Phys., 103, 5979 (1995)
[27] Sokolovski, D.; Castillo, J. F.; Tully, C., Semiclassical angular scattering in the \(F + H_2\) → HF + H reaction: Regge pole analysis using the Padé approximation, Chem. Phys. Lett., 313, 225 (1999)
[28] Sokolovski, D.; Castillo, J. F., Differential cross sections and Regge trajectories for the \(F + H_2\) → HF + H reaction, Chem. Phys. Lett., 2, 507 (2000)
[29] Sokolovski, D.; Sen, S. K.; Aquilanti, V.; Cavalli, S.; De Fazio, D., Interacting resonances in the \(F + H_2\) reaction revisited: Complex terms, Riemann surfaces, and angular distributions, J. Chem. Phys., 126, 084305 (2007)
[30] Sokolovski, D.; De Fazio, D.; Cavalli, S.; Aquilanti, V., Overlapping resonances and Regge oscillations in the state-to-state integral cross sections of the \(F + H_2\) reaction, J. Chem. Phys., 126, 12110 (2007)
[31] Sokolovski, D.; De Fazio, D.; Cavalli, S.; Aquilanti, V., On the origin of the forward peak and backward oscillations in the \(F + H_2(v = 0) \to HF(v^\prime = 2) + H\) reaction, Phys. Chem. Chem. Phys., 9, 1 (2007)
[32] Aoiz, F. J.; Bañares, L.; Castillo, J. F.; Sokolovski, D., Energy dependence of forward scattering in the differential cross section of the \(H + D_2 \to HD(v^\prime = 3, j^\prime = 0) + D\) reaction, J. Chem. Phys., 117, 2546 (2002)
[33] Sokolovski, D., Glory and thresholds effects in \(H + D_2\) reactive angular scattering, Chem. Phys. Lett., 370, 805 (2003)
[34] Sokolovski, D., Complex-angular-momentum analysis of atom-diatom angular scattering: Zeros and poles of the \(S\) matrix, Phys. Rev. A, 62 (2000), 024702-01
[35] Sokolovski, D.; Msezane, A. Z.; Felfli, Z.; Ovchinnikov, S. Yu.; Macek, J. H., What can one do with Regge poles?, Nucl. Instrum. Methods Phys. Res., B Beam Interact. Mater. Atoms, 261, 133 (2007)
[36] Totenhofer, A. J.; Noli, C.; Connor, J. N.L., Dynamics of the I + HI → IH + I reaction: Application of nearside-farside, local angular momentum and resummation theories using the Fuller and Hatchell decompositions, Phys. Chem. Chem. Phys., 12, 8772-8791 (2010)
[37] Macek, J. H.; Krstic, P. S.; Ovchinnikov, S. Yu., Regge oscillations in integral cross sections for proton impact on atomic hydrogen, Phys. Rev. Lett., 93, 183203 (2004)
[38] Sokolovski, D., Complex-angular-momentum (CAM) route to reactive scattering resonances: from a simple model to the \(F + H_2\) → HF + H reaction, Phys. Scr., 78, 058118 (2008)
[39] Burke, P. G.; Tate, C., A program for calculating Regge trajectories in potential scattering, Comput. Phys. Commun., 1, 97 (1969)
[40] Sokolovski, D.; Felfli, Z.; Ovchinnikov, S. Yu.; Macek, J. H.; Msezane, A. Z., Regge oscillations in electron-atom elastic cross sections, Phys. Rev. A, 76, 012705 (2007)
[41] Bessis, D.; Haffad, A.; Msezane, A. Z., Momentum-transfer dispersion relations for electron-atom cross sections, Phys. Rev. A, 49, 3366 (1994)
[42] Baker, G. A., The Essentials of Padé Approximations (1975), Academic: Academic New York
[43] Petkovic, M. S.; Carstensen, C.; Trajkovic, M., Weierstrass formula and zero-finding methods, Numer. Math., 69 (1995) · Zbl 0821.65028
[44] Bailey, D. H., Algorithm 719, Multiprecision translation and execution of Fortran programs, ACM Trans. Math. Softw., 19, 3, 288 (1993) · Zbl 0889.68015
[45] Numerical Algorithms Group, Fortran Library Manual, Mark 19, subroutine G05CAF, NAG, OXFORD, 2002.; Numerical Algorithms Group, Fortran Library Manual, Mark 19, subroutine G05CAF, NAG, OXFORD, 2002.
[46] Numerical Algorithms Group, Fortran Library Manual, Mark 21, subroutine C02AFF, NAG, OXFORD, 2004.; Numerical Algorithms Group, Fortran Library Manual, Mark 21, subroutine C02AFF, NAG, OXFORD, 2004.
[47] Numerical Algorithms Group, Fortran Library Manual, Mark 21, subroutine E02ACF, NAG, OXFORD, 2004.; Numerical Algorithms Group, Fortran Library Manual, Mark 21, subroutine E02ACF, NAG, OXFORD, 2004.
[48] Olson, R. E.; Smith, F. T., Phys. Rev. A, 6, 526 (1972), (Erratum)
[49] K.-E. Thylwe, unpublished.; K.-E. Thylwe, unpublished.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.