A Fourier-Lanczos method for calculating energy levels without storing or calculating matrices

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Abstract
A variational method for calculating vibrational energy levels of polyatomic molecules is developed and applied. Using this new approach energy levels (and wavefunctions) may be determined without storing Hamiltonian matrix elements. An explicit finite representation of the Hamiltonian is circumvented by representing basis functions on an evenly spaced grid, employing Fourier transforms to evaluate the action of the Hamiltonian operator on the basis functions, and using the Lanczos algorithm to calculate eigenvalues. Unhampered by the need to store a finite representation of the Hamiltonian, the method should make it possible to calculate many energy levels of molecules with many vibrational degrees of freedom.