Wavelets in curvilinear coordinate quantum calculations: H2+ electronic states

Abstract

A low-storage filter diagonalization method for quantum eigenenergy calculation or for spectral analysis of time signals Multiscale wavelets are used to solve the quantum eigenvalue equations for the hydrogen molecular ion H 2 ϩ in the Born-Oppenheimer approximation. Normally restricted to Cartesian systems, ''wavelets on the interval'' ͑a normal wavelet family augmented by special edge functions͒ have recently been applied to such boundary value problems as the hydrogen atom in spherical polar coordinates ͓J. Mackey, J. L. Kinsey, and B. R. Johnson, J. Comp. Phys. 168, 356 ͑2001͔͒. These methods are extended here to ground and excited electronic states of the simplest molecule, for which the electronic Hamiltonian is separable in confocal elliptic coordinates. The set of curvilinear coordinate quantum systems for which wavelet bases have been applied is thus enlarged.

Multiresolution quantum chemistry: Basic theory and initial applications

“…20,55,56 The results for H 2 ͑bond length 1.4 bohrs, box size Lϭ89.6 bohrs) are given in Table I. In contrast to the recent three-dimensional, mixed-basis results of Pahl and Handy, 55 no extrapolation was necessary and the best result is accurate to about 10 Ϫ10 a.u.…”

“…Previous applications of wavelets and multiresolution analysis to quantum chemistry [19][20][21][22][23][24][25][26][27][28] have almost exclusively employed single-component smooth wavelets and have primarily focused upon periodic systems with pseudopotentials. Prior investigations employing multiwavelets for electronic structure have been limited in scope.…”

“…Prior investigations employing multiwavelets for electronic structure have been limited in scope. 20,29 Limited applications of wavelets have been made to molecules including all electrons, and the largest prior such calculation that we are aware of is to the oxygen molecule 21 employing a large unit cell in a periodic code. More extensive application has been made of multigrid approaches 4,[30][31][32] and, most significantly, the numerical approach of Becke.…”

Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. III. Representations and calculations

“…We comment, however, that the development of affine and Wilson-Daubechies-type wavelet technologies, for use with non-Cartesian coordinates in the molecular physics context, is currently an active area of investigation. [32][33][34] In any event, we will for the moment assume a Hamiltonian Weyl symbol ͑Paper II͒ of the standard form…”

Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. II. Construction and optimization

“…The development of affine wavelet technologies, for use with non-Cartesian coordinates in the molecular physics context, is currently an active area of investigation, notably by Johnson and co-workers. 45,46 The development of Wilson-type wavelets on the interval is also being investigated. 28 Before moving on, we note that the phase space picture adopted here affords a natural view of the deep interrelation between weylets and coherent states.…”