B. R. Johnson

New numerical methods applied to solving the one‐dimensional eigenvalue problem

Two new numerical methods, the log derivative and the renormalized Numerov, are developed and applied to the calculation of bound‐state solutions of the one‐dimensional Schroedinger equation. They are efficient and stable; no convergence difficulties are encountered with double minimum potentials. A useful interpolation formula for calculating eigenfunctions at nongrid points is also derived. Results of example calculations are presented and discussed.

The renormalized Numerov method applied to calculating bound states of the coupled‐channel Schroedinger equation

The renormalized Numerov method, which was recently developed and applied to the one‐dimensional bound state problem [B. R. Johnson, J. Chem. Phys. 67, 4086 (1977)], has been generalized to compute bound states of the coupled‐channel Schroedinger equation. Included in this presentation is a generalization of the concept of a wavefunction node and a method for detecting these nodes. By utilizing node count information it is possible to converge to any specific eigenvalue without the need of an initial close guess and also to calculate degenerate eigenvalues and determine their degree of degeneracy. A useful interpolation formula for calculating the eigenfunctions at nongrid points is also given. Results of example calculations are presented and discussed. One of the example problems is the single center expansion calculation of the 1sσg and 2sσg states of H+2.

On hyperspherical coordinates and mapping the internal configurations of a three body system

A procedure for mapping the internal configurations of a three body system to points in a three dimensional ’’configuration space’’ is presented. The mapping is based on a modified version of the Smith–Whitten symmetric hyperspherical coordinate system. Several years ago, Kuppermann developed a mapping procedure based on a different hyperspherical coordinate system. Transformation equations relating these two mappings are derived. A comparison shows that the present method produces maps identical to Kuppermann’s method. The coordinate axes in configuration space, however, are rotated by 90 degrees so that the z axis is coincident with the axis of kinematic rotation.

The classical dynamics of three particles in hyperspherical coordinates

Classical dynamics of the three body problem, formulated in hyperspherical coordinates, is investigated. Hamilton’s equations of motion are derived and then reduced from 12th order to eighth order. In addition to the general case in which the system moves in three‐dimensional space, the special cases of planar motion, collinear motion, and zero angular momentum motion are studied and related. A brief description of the theory of small amplitude vibrations and normal modes in hyperspherical coordinates is also presented.

The quantum dynamics of three particles in hyperspherical coordinates

A derivation of the quantum mechanical wave equation for the three body problem expressed in hyperspherical coordinates is presented. The coordinates, due to Smith and Whitten and later modified by Johnson [B. R. Johnson, J. Chem. Phys. 73, 5051 (1980)], are used in this study. The analysis is presented from a point of view that emphasizes the role of the three‐dimensional configuration space that is associated with these coordinates. Three types of motion are analyzed: full three dimensional; motion restricted to a plane surface; and zero angular momentum motion.

Classical Trajectory Study of the Effect of Vibrational Energy on the Reaction of Molecular Hydrogen with Atomic Oxygen.

The dynamics of the reaction O+H2(v) →OH+H is studied by means of three dimensional classical trajectory calculations on an LEPS potential energy surface. Rate constants are calculated for the two cases in which the H2 molecule is initially in the v=0 and v=1 vibrational state. In the temperature range 298–1000 °K these rates are fit very well by the formulas (cm3 molecule−1 sec−1) k=2.81T×10−14 exp(−4279/T) and k=4.65T×10−14 exp(−1868/T). The calculated value of k at 300 °K is 2.8×10−14 cm3 molecule−1 sec−1 which is below the upper bound established by Birely [J. H. Birely, J.V.V. Kasper, F. Hai, and L. A. Darnton, Chem. Phys. Lett. 31, 220 (1975)]. The branching ratio Γ, defined as the ratio of the rates for populating the v′=1 and v′=0 state of OH when H2 is initially in the v=1 state is also calculated and fit by the expression Γ=2.3 exp(196/T). The value at 300 °K is 4.4.

On the adiabatic invariance method of calculating semiclassical eigenvalues

In this paper we analyze and further develop the adiabatic invariance method for computing semiclassical eigenvalues. This method, which was recently introduced by Solov’ev, is basically an application of the Ehrenfest adiabatic hypothesis. The eigenvalues are determined from a classical calculation of the energy as the time dependent Hamiltonian H(t)=H0+s(t)H1 is switched adiabatically from the separable reference Hamiltonian H0 to the system Hamiltonian H0+H1. A systematic study is carried out to determine the best form for the switching function, s(t), to maximize the rate of convergence of the energy to its adiabatic limit. Five switching functions, including the linear function used by Solov’ev, are defined and tested on three different systems. The linear function is found to have a very slow convergence rate compared to the others. The classical energy is shown to be a periodic function of the angle coordinates of H0. The coefficients of the Fourier series representation of this function are then s...

A quantum mechanical investigation of vibrational energy transfer in O(3P)+H2O collisions

Cross sections for the vibrational excitation of H2O in collision with O(3P) are calculated for relative collision energies of 0.5 to 3.0 eV by the vibrational close‐coupling rotational infinite order sudden method using an accurate potential energy surface. The excitation cross sections obtained by this quantum mechanical calculation are compared to results of a recently published quasiclassical trajectory study which used the same potential surface. Very large differences between the quantum mechanical and classical trajectory results are found.

Semiclassical vibrational eigenvalues of H+3, D+3, and T+3 by the adiabatic switching method

The adiabatic switching method is used to calculate the semiclassical vibrational eigenvalues of H+3, D+3, and T+3. The results are in good agreement with quantum calculations and are more accurate than previous semiclassical results. The calculations were facilitated by the use of hyperspherical coordinates.

Comment on the Truhlar–Horowitz functional representation of the H+H2 potential surface

The Truhlar‐Horowitz H3 potentials subroutine is commented upon. It is contended that the TH functional representation has discontinuous derivatives for certain isosceles triangle configurations. (AIP).