The development of hyperspherical techniques has been crucial in the study of the dynamics of elementary chemical reactions in the gas phase. During the nineties, thanks to these advances some different algorithms capable to solve exactly the time independent Schrodinger equation for atom-diatom reactions have been developed [1-3]. Their application has produced in the years a wide set of benchmark data. For systems involving many atoms, these codes have been very useful to test approximated theoretical methodologies necessary to study the kinetics of applied chemical environments and to understand the behaviour of reactive observables measured in cross beam experiments or simulated in large scale calculations. In the hyperquantization algorithm [1], developed in our laboratory, we have used an expansion in hyperspherical harmonics to solve the eigenvalue problem at fixed radius, in a similar way as the quantum solution of the electronic levels of the hydrogen atom are solved in analytical theories. Unfortunately, the hyperspherical harmonics expansion converges slowly and a huge number of harmonics with large quantum numbers is required making the code largely inefficient. However, in the hyperquantization algorithm [1] this problem is by-passed by using in place of hyperspherical harmonics their discrete analogous that approximate very well the values of the hyperspherical harmonics over a grid of points when large values of the quantum numbers are involved [4]. The main advantage to use orthogonal polynomials of a discrete variable, namely the Hahn polynomials, is that exploiting the algebra of the hyperangular moments large sums among these quantities can be done analytically so that the explicit calculation of the harmonics or of the Hahn polynomials is actually not required in practice making the scattering algorithm very fast and efficient. Moreover, the mathematical properties of the Hahn polynomials permit to formulate the eigenvalue problem in terms of matrices with particular structures efficiently exploited by linear algebra methods [5]. Theoretical details of the hyperquantization algorithm and its differences with the other available methods will be presented at the conference. Results of benchmark studies carried out with this methodology in the last ten years will be also shown and discussed. [1] V. Aquilanti, S. Cavalli and D. De Fazio. 'Hypherquantization algorithm. I. Theory for triatomic systems'. J. Chem. Phys., 109, (1998) 3792-3804. [2] J.M. Launay and M. Ledourneuf. 'Hyperspherical close-coupling calculation of integral cross-sections for the reaction H + H2 -> H2 + H'. Chem. Phys. Lett. 163 (1989) 178-188. [3] D. Skouteris, J.F. Castillo and D.E. Manolopoulos. 'ABC: a quantum reactive scattering program'. Comput. Phys. Commun., 133, (2000) 128-135. [4] V. Aquilanti, S. Cavalli and D. De Fazio; 'Angular and Hyperangular momentum coupling coefficients as Hahn Polynomials'. J. Phys. Chem., 99, (1995) 15694-15698. [5] V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi, A. Aguilar, X. Gimenez and J. M. Lucas; `Hypherquantization algorithm. II. Implementation for the F + H2 reaction dynamics including open-shell and spin-orbit interaction'. J. Chem. Phys., 109, (1998) 3805-3818.
Discretized Harmonics for Large Angular Momentum Reactivity
Dario De Fazio;
2017
Abstract
The development of hyperspherical techniques has been crucial in the study of the dynamics of elementary chemical reactions in the gas phase. During the nineties, thanks to these advances some different algorithms capable to solve exactly the time independent Schrodinger equation for atom-diatom reactions have been developed [1-3]. Their application has produced in the years a wide set of benchmark data. For systems involving many atoms, these codes have been very useful to test approximated theoretical methodologies necessary to study the kinetics of applied chemical environments and to understand the behaviour of reactive observables measured in cross beam experiments or simulated in large scale calculations. In the hyperquantization algorithm [1], developed in our laboratory, we have used an expansion in hyperspherical harmonics to solve the eigenvalue problem at fixed radius, in a similar way as the quantum solution of the electronic levels of the hydrogen atom are solved in analytical theories. Unfortunately, the hyperspherical harmonics expansion converges slowly and a huge number of harmonics with large quantum numbers is required making the code largely inefficient. However, in the hyperquantization algorithm [1] this problem is by-passed by using in place of hyperspherical harmonics their discrete analogous that approximate very well the values of the hyperspherical harmonics over a grid of points when large values of the quantum numbers are involved [4]. The main advantage to use orthogonal polynomials of a discrete variable, namely the Hahn polynomials, is that exploiting the algebra of the hyperangular moments large sums among these quantities can be done analytically so that the explicit calculation of the harmonics or of the Hahn polynomials is actually not required in practice making the scattering algorithm very fast and efficient. Moreover, the mathematical properties of the Hahn polynomials permit to formulate the eigenvalue problem in terms of matrices with particular structures efficiently exploited by linear algebra methods [5]. Theoretical details of the hyperquantization algorithm and its differences with the other available methods will be presented at the conference. Results of benchmark studies carried out with this methodology in the last ten years will be also shown and discussed. [1] V. Aquilanti, S. Cavalli and D. De Fazio. 'Hypherquantization algorithm. I. Theory for triatomic systems'. J. Chem. Phys., 109, (1998) 3792-3804. [2] J.M. Launay and M. Ledourneuf. 'Hyperspherical close-coupling calculation of integral cross-sections for the reaction H + H2 -> H2 + H'. Chem. Phys. Lett. 163 (1989) 178-188. [3] D. Skouteris, J.F. Castillo and D.E. Manolopoulos. 'ABC: a quantum reactive scattering program'. Comput. Phys. Commun., 133, (2000) 128-135. [4] V. Aquilanti, S. Cavalli and D. De Fazio; 'Angular and Hyperangular momentum coupling coefficients as Hahn Polynomials'. J. Phys. Chem., 99, (1995) 15694-15698. [5] V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi, A. Aguilar, X. Gimenez and J. M. Lucas; `Hypherquantization algorithm. II. Implementation for the F + H2 reaction dynamics including open-shell and spin-orbit interaction'. J. Chem. Phys., 109, (1998) 3805-3818.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
