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Variational basis-set formulations of the quantum mechanical reactive scattering problem lead to large, dense sets of equations. In previous work, we showed that the generalized minimal residual (GMRes) algorithm is sometimes competitive in terms of computer time with direct methods for these dense matrices, even when complex-valued boundary conditions are used, leading to non-Hermitian matrices. This paper presents a preconditioning scheme to accelerate convergence and improve performance. We block the potential energy coupling into a series of distortion blocks, and we employ the outgoing wave variational principle with nonorthogonal basis functions, including both dynamically adapted Green’s functions for the distortion blocks and also square integrable functions. The coefficient matrix of the resulting linear system couples the blocks. We have found that preconditioners formed from diagonal blocks of the coefficient matrix corresponding to the distortion blocks and vibrational blocks are effective at accelerating the iterative method in every test case, by factors of 2.9–20, with an average speedup of a factor of 6.5. The storage requirements and computational efficiency of the new scheme compare favorably to those for preconditioners based on banded matrices of variable bandwidth. The new preconditioners yield converged transition probabilities in less computer time than a direct solver even in cases which do not converge in a reasonable amount of time without preconditioning, and the average speedup compared to the direct solution is a factor of 7.6. [ABSTRACT FROM AUTHOR]

A complex GMRes (generalized minimum residual) algorithm is presented and used to solve dense systems of linear equations arising in variational basis-set approaches to quantum-mechanical reactive scattering. The examples presented correspond to physical solutions of the Schrödinger equation for the reactions O+HD→OH+D, D+H2→HD+H, and H+H2→H2+H. It is shown that the computational effort for solution with GMRes depends upon both the dimension of the linear system and the total energy of the reaction. In several cases with dimensions in the range 1110–5632, GMRes outperforms the LAPACK direct solver, with speedups for the linear equation solution as large as a factor of 23. In other cases, the iterative algorithm does not converge within a reasonable time. These convergence differences can be correlated with ‘‘indices of diagonal dominance,’’ which we define in detail and which are relatively easy to compute. Furthermore, we find that for a given energy, the computational effort for GMRes can vary with dimension as favorably as M1.7, where M is the dimension of the linear system, whereas the computer time for a direct method is approximately proportional to the cube of the dimension of the linear system. [ABSTRACT FROM AUTHOR]

In attempting to understand the ground vibronic state of CH+4, a new formalism was developed for discussing molecular symmetry groups. Various diabatic basis sets and approximations for Jahn–Teller vibronic problems were examined, using the relatively simple Exe case. Finally, the symmetries and form of the ground vibronic states of CH+4 were produced. [ABSTRACT FROM AUTHOR]