An R matrix approach to the solution of coupled equations for atom–molecule reactive scattering
15 Nov 1976-Journal of Chemical Physics (American Institute of Physics)-Vol. 65, Iss: 10, pp 4272-4282
TL;DR: In this paper, a new method for solving the coupled linear differential equations which appear in the theoretical statement of many types of scattering phenomena is described, combining the speed advantage of propagation methods with the inherent stability of R • matrix methods.
Abstract: We describe in detail a new method for solving the coupled linear differential equations which appear in the theoretical statement of many types of scattering phenomena. Combining the speed advantage of propagation methods with the inherent stability of R‐matrix methods, this method is fast and is unaffected by exponential growth of closed channels. We describe the propagation of the R‐matrix in terms of a collinear reactive scattering problem, and provide numerical results for several systems, clearly demonstrating the speed, stability, and accuracy of this method.
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TL;DR: In this paper, a new method is presented for the solution of the time dependent SchrBdinger equation in its application to physical and chemical molecular phenomena, which is based on discretizing space and time on a grid, and using the Fourier method to produce both spatial derivatives, and second order differencing for time derivatives.
1,138 citations
TL;DR: RMATRX1 as mentioned in this paper is a general program to calculate atomic continuum processes using the R -matrix method, including electronatom and electron-ion scattering, and radiative processes such as bound-bound transitions, photoionization and polarizabilities.
679 citations
TL;DR: The theory of reactive rearrangement scattering for three atoms in three dimensions using adiabatically adjusting, principal axes hyperspherical (APH) coordinates is given in this article.
Abstract: The theory of reactive (rearrangement) scattering for three atoms in three physical dimensions using adiabatically adjusting, principal axes hyperspherical (APH) coordinates is given. The relationships of the APH coordinates to Delves and Jacobi coordinates are given, and the kinetic energy operator is shown to be relatively simple. Procedures for solving the equations via either an exact coupled channel (CC) method or an optimum centrifugal sudden (CSAPH) approximation are given as well as procedures for applying scattering boundary conditions. Surface functions of two angles are obtained using a finite element method with an optimized, nonuniform mesh, and the CC equations are solved using the efficient VIVAS method. Sample CC results are given for the H3 system. The present approach has the advantages that all arrangements are treated fully equivalently; it is a principal axis system, so that both axes and internal coordinates swing smoothly with the reactions; it is directly applicable to both symmetric and unsymmetric systems and mass combinations and all total angular momenta; it gives convenient mappings for visualization of potential energy surfaces and wave functions; only regular radial solutions are required; all coordinate matching is by simple projection; and the expensive parts of the calculation are energy independent, so that, once they are done, the scattering matrices can be rapidly generated at the large numbers of energies needed to map out reactive thresholds and resonances. Accurate reactive scattering calculations are now possible for many chemically interesting reactions that were previously intractable.
531 citations
TL;DR: The current state of the theory and its application to low-energy electron-molecule collisions is reviewed in this paper, where the emphasis is on elastic scattering and vibrational and rotational excitation of small diatomic and polyatomic molecules.
Abstract: The current state of the theory and its application to low-energy electron-molecule collisions is reviewed. The emphasis is on elastic scattering and vibrational and rotational excitation of small diatomic and polyatomic molecules. New and traditional theoretical approaches are described, and the results of calculations are compared with existing experimental measurements.
520 citations
TL;DR: The R-matrix method is an embedding procedure which is based on the division of space into an inner region where the physics is complicated and an outer region for which greatly simplified equations can be solved.
372 citations
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1,757 citations
TL;DR: In this paper, the energy dependence of the reaction and scattering cross sections is analyzed in terms of the collision matrix, which is independent of the size of the internal region and the number of angular momenta of the particles.
Abstract: The results of two preceding articles on the energy dependence of the reaction and scattering cross sections are generalized. As before, the configuration space is divided into two parts. In the internal region or reaction zone there is no restriction on the type of interaction between the particles. In the external region, on the other hand, the interaction takes place between the colliding or separating particles without changing their structure (wave function). The present article deals with a more general situation than the preceding ones, first by allowing for an interaction, although only of a restricted type, outside the reaction zone. The most common type of interaction which plays a role outside the reaction zone is the electrostatic interaction of the colliding or separating particles. In addition, the present article does not restrict the angular momentum of the particles to zero, but permits the treatment of arbitrary angular momenta. The cross sections are expressed in well-known fashion in terms of the collision matrix $\mathfrak{U}$ which is, of course, independent of the size of the internal region. $\mathfrak{U}$, in its turn, is expressed by the $\mathfrak{R}$ matrix ((35) and (38)) and the quantities $\ensuremath{\omega}$, $\mathfrak{B}$, and $\mathfrak{C}$ describing the interaction in the external region. $\mathfrak{R}$ as function of energy is given by (24), and a number of properties of this function are enumerated. None of the quantities $\mathfrak{R}$, $\ensuremath{\omega}$, $\mathfrak{B}$, and $\mathfrak{C}$ are strictly independent of the size of the internal region, although the combination (38), i.e., the collision matrix, of course, is.
779 citations
TL;DR: In this article, an analytic semi-empirical expression for the ground-state potential energy surface of the H3 system was developed using the valence-bond formulation with the inclusion of overlap and three center terms.
Abstract: An analytic semiempirical expression is developed for the ground‐state potential‐energy surface of the H3 system. Use is made of the valence‐bond formulation with the inclusion of overlap and three‐center terms. The diatomic Coulomb and exchange integrals are estimated from accurate values for the H2 molecule by a modified London—Eyring—Sato procedure, while the three‐center integrals are approximated by simple formulas. A linear, symmetric saddle‐point configuration of minimum energy is found that has properties in approximate agreement, though not identical, with other estimates from theory and experimental rate data. The details of the surface are presented in terms of contour maps for a variety of distances and angles. Because of the inclusion of overlap and three‐center terms, a more realistic energy is expected for the nonlinear configurations than could be obtained from previous potential energy formulas. The utility of the present method for dynamical studies of reaction cross sections is indicated.
610 citations
TL;DR: In this article, a new method was developed for integrating coupled differential equations arising in bound state and scattering problems in quantum mechanics, and wavefunctions were easily constructed in piecewise analytic form, to any prescribed accuracy.
Abstract: A new method is developed for integrating coupled differential equations arising in bound state and scattering problems in quantum mechanics. The wavefunctions are easily constructed in piecewise analytic form, to any prescribed accuracy.
531 citations
TL;DR: In this article, a semi-empirical formula for computing quantum-mechanical transition probabilities for collinear collision of an atom with a diatomic molecule is given.
Abstract: Exact quantum‐mechanical calculations of the transition probabilities for the collinear collision of an atom with a diatomic molecule are performed. The diatomic molecule is treated as a harmonic oscillator. A range of interaction potentials from very hard to very soft are considered. It is found that for ``realistic'' interaction potentials the approximate calculations of Jackson and Mott are consistently high, even when the transition probabilities are low and good approximate results are expected. In some cases double and even triple quantum jumps are more important than single quantum jumps. Comparisons are made with exact classical calculations. A semiempirical formula is given for computing quantum‐mechanical transition probabilities from classical calculations.
456 citations