Adiabatic pseudospectral methods for multidimensional vibrational potentials

Abstract

Articles you may be interested inA multidimensional pseudospectral method for optimal control of quantum ensembles A classical determination of vibrationally adiabatic barriers and wells of a collinear potential energy surfaceWe describe a new algorithm for computing eigenvalues, spectral intensities, and selected eigenvectors of multidimensional vibrational potential surfaces. The method involves a synthesis of pseudospectral and sequential adiabatic reduction methods and merges the storage and computational advantages of the former with the improved basis set generated by the latter. The recursive residue generation method, which utilizes a Lanczos-based diagonalization procedure, is employed to calculate the observables. As a test case, we apply the method to computation of the infrared and stimulated emission pumping spectra for the HCN molecule and demonstrate a very large (one to three orders of magnitude) reduction in computational effort (for comparable accuracy) as compared to discrete variable representation (DVR)/adiabatic reduction or standard collocation approaches. We expect that this advantage will be increased considerably for larger (e.g., tetra-atomic) systems and will permit accurate basis set calculations on such systems to be carried out in a straightforward fashion.324

Numerical Analysis of the Adiabatic Variable Method for the Approximation of the Nuclear Hamiltonian

“…We are then lead to search for methods allowing us to further reduce the number of basis functions. The pseudo-spectral adiabatic variable method proposed in [9,12] is one such pertinent discretization tool that seems to give quite good results in practice. Its principle is presented below for a triatomic molecule.…”

“…Let the Laplace operator be written in Jacobi coordinates (R, r, θ) (cf. [9]), and let us assume that we want to find a function ψ on the open brick 1 …”

“…[1,[9][10][11][12]16]) consists in the evaluation of the eigenmodes of some Hamiltonian operator corresponding to eigenvalues smaller than some prescribed value E MAX .…”

“…By tensorization the computation (58) can be done in c max(M, N ) 5 operations, less than the number of operations required by the computation of ψ δ (for instance, the diagonalization of 2D Hamiltonians is of higher complexity) [1,9,12]. Indeed, our goal is to compute for {(p , q , j); |Λ p ,q ,j | ≥ (1 + )E MAX } the term:…”

Dissociative chemisorption of H2 on Cu(100): A four-dimensional study of the effect of parallel translational motion on the reaction dynamics

“…At the same time, it is known that the Lanczos algorithm converges specific eigenvalues more quickly if the vector used to start the Lanczos recursion significantly overlaps the corresponding eigenvectors ͑''guided Lanczos''͒. [29][30][31][32] The above two considerations lead to an obvious choice of starting vector: For a single initial state calculation, one should select ⌿(T), because it significantly overlaps the resonance states of interest ͓see Eq. ͑1͔͒ and because it is available at no extra cost from the first part of the calculation.…”