^{1}and Randall S. Dumont

^{1,a)}

### Abstract

This article presents a new complex absorbing potential (CAP) block Lanczos method for computing scattering eigenfunctions and reaction probabilities. The method reduces the problem of computing energy eigenfunctions to solving two energy dependent systems of equations. An energy independent block Lanczos factorization casts the system into a block tridiagonal form, which can be solved very efficiently for all energies. We show that CAP-Lanczos methods exhibit instability due to the non-normality of CAP Hamiltonians and may break down for some systems. The instability is not due to loss of orthogonality but to non-normality of the Hamiltonian matrix. While use of a Woods–Saxon exponential CAP—as opposed to a polynomial CAP—reduced non-normality, it did not always ensure convergence. Our results indicate that the Arnoldi algorithm is more robust for non-normal systems and less prone to break down. An Arnoldi version of our method is applied to a nonadiabatictunneling Hamiltonian with excellent results, while the Lanczos algorithm breaks down for this system.

We would like to thank Tucker Carrington, Jr. for helpful discussions on non-normal matrices and the Natural Sciences and Engineering Research Council of Canada for financial support.

I. INTRODUCTION

II. CAP-BLOCK LANCZOS METHOD

A. The matrix

B. Block Lanczos iteration

C. Collinear

III. INVESTIGATION OF CAP-LANCZOS AND CAP-ARNOLDI FOR NON-NORMAL MATRICES

A. Nonadiabatictunneling model

B. Collinear ClHCl Hamiltonian

IV. CAP-ARNOLDI SCATTERING

V. CONCLUSION

### Key Topics

- Eigenvalues
- 63.0
- Tunneling
- 12.0
- Non adiabatic reactions
- 11.0
- Hydrogen reactions
- 9.0
- Subspaces
- 7.0

## Figures

State to state reaction probabilities for the collinear reaction below .

State to state reaction probabilities for the collinear reaction below .

Log plot of (a) the components of the intermediate solution of the decomposition of Eq. (13) and (b) the residual error vs size of Krylov subspace, for , 1.25, and for the collinear reaction.

Log plot of (a) the components of the intermediate solution of the decomposition of Eq. (13) and (b) the residual error vs size of Krylov subspace, for , 1.25, and for the collinear reaction.

Log plot of the residual error (solid line) after 6500 block Lanczos iterations (for energies below ) for the collinear reaction. The dotted line corresponds to the last element of intermediate solution.

Log plot of the residual error (solid line) after 6500 block Lanczos iterations (for energies below ) for the collinear reaction. The dotted line corresponds to the last element of intermediate solution.

Plot of the number Lanczos vectors (twice the number of block Lanczos iterations) required to converge the solution of the system of equations with a residual error of less than . The results are for 4000 energies below . The dots show those eigenvalues (real part) of the colllinear Hamiltonian that have also converged with a residual error less than .

Plot of the number Lanczos vectors (twice the number of block Lanczos iterations) required to converge the solution of the system of equations with a residual error of less than . The results are for 4000 energies below . The dots show those eigenvalues (real part) of the colllinear Hamiltonian that have also converged with a residual error less than .

Diabatic and adiabatic potential energy curves for the nonadiabatic tunneling Hamiltonian.

Diabatic and adiabatic potential energy curves for the nonadiabatic tunneling Hamiltonian.

Log plot of the eigenvector of the Lanczos tridiagonal matrix corresponding to the lowest eigenvalue of the nonadiabatic tunneling quartic CAP Hamiltonian for (a) , (b) , (c) , and (d) . The dotted lines correspond to associated spurious Ritz values.

Log plot of the eigenvector of the Lanczos tridiagonal matrix corresponding to the lowest eigenvalue of the nonadiabatic tunneling quartic CAP Hamiltonian for (a) , (b) , (c) , and (d) . The dotted lines correspond to associated spurious Ritz values.

Log plot of the eigenvector of the Arnoldi Hessenberg matrix corresponding to the lowest eigenvalue of the nonadiabatic tunneling CAP Hamiltonian with .

Log plot of the eigenvector of the Arnoldi Hessenberg matrix corresponding to the lowest eigenvalue of the nonadiabatic tunneling CAP Hamiltonian with .

Log plot of the eigenvector of the Lanczos tridiagonal matrix corresponding to the lowest eigenvalue of the nonadiabatic tunneling Woods–Saxon CAP Hamiltonian, .

Log plot of the eigenvector of the Lanczos tridiagonal matrix corresponding to the lowest eigenvalue of the nonadiabatic tunneling Woods–Saxon CAP Hamiltonian, .

Log plot of the eigenvectors of the Lanczos tridiagonal matrix corresponding to (a) and (b) for the colllinear Hamiltonian. The dotted lines correspond to spurious Ritz values.

Log plot of the eigenvectors of the Lanczos tridiagonal matrix corresponding to (a) and (b) for the colllinear Hamiltonian. The dotted lines correspond to spurious Ritz values.

Log plot of the eigenvector of the Arnoldi Hessenberg matrix corresponding to (a) and (b) for the colllinear Hamiltonian.

Log plot of the eigenvector of the Arnoldi Hessenberg matrix corresponding to (a) and (b) for the colllinear Hamiltonian.

Log plot of reflection (solid line) and transmission (dot line) probabilities for the nonadiabatic tunneling Hamiltonian. Thirteen reflection resonances are evident at which we have a reflection probability of 100%.

Log plot of reflection (solid line) and transmission (dot line) probabilities for the nonadiabatic tunneling Hamiltonian. Thirteen reflection resonances are evident at which we have a reflection probability of 100%.

Log plot of (a) the residual error for , 1.25, and and (b) the residual error after 20 000 block Lanczos iterations for the collinear reaction.

Log plot of (a) the residual error for , 1.25, and and (b) the residual error after 20 000 block Lanczos iterations for the collinear reaction.

## Tables

State to state reaction probabilities for the collinear reaction.

State to state reaction probabilities for the collinear reaction.

Condition number for the nonadiabatic tunneling Hamiltonian augmented with a quartic CAP for different CAP strengths.

Condition number for the nonadiabatic tunneling Hamiltonian augmented with a quartic CAP for different CAP strengths.

Residual and relative errors for the lowest eigenvalues obtained by the Lanczos algorithm for the nonadiabatic tunneling quartic CAP Hamiltonian with different CAP strengths. The eigenvalues obtained by diagonalizing the Hamiltonian with IMSL are considered as exact. Only the real part of the eigenvalues and Ritz values are reported.

Residual and relative errors for the lowest eigenvalues obtained by the Lanczos algorithm for the nonadiabatic tunneling quartic CAP Hamiltonian with different CAP strengths. The eigenvalues obtained by diagonalizing the Hamiltonian with IMSL are considered as exact. Only the real part of the eigenvalues and Ritz values are reported.

Relative errors for the lowest eigenvalues obtained by the Arnoldi algorithm for the nonadiabatic tunneling quartic CAP Hamiltonian with different CAP strengths. The eigenvalues obtained by diagonalizing the Hamiltonian with IMSL are considered as exact. Only the real part of the eigenvalues and Ritz values are reported.

Relative errors for the lowest eigenvalues obtained by the Arnoldi algorithm for the nonadiabatic tunneling quartic CAP Hamiltonian with different CAP strengths. The eigenvalues obtained by diagonalizing the Hamiltonian with IMSL are considered as exact. Only the real part of the eigenvalues and Ritz values are reported.

Relative errors for eigenvalues and obtained by the Arnoldi and Lanczos algorithms for the ClHCl system augmented with a Woods–Saxon CAP. The eigenvalues obtained by diagonalizing the Hamiltonian with IMSL are considered as exact. Only the real part of the eigenvalues and Ritz values are reported.

Relative errors for eigenvalues and obtained by the Arnoldi and Lanczos algorithms for the ClHCl system augmented with a Woods–Saxon CAP. The eigenvalues obtained by diagonalizing the Hamiltonian with IMSL are considered as exact. Only the real part of the eigenvalues and Ritz values are reported.

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