Circumventing problems introduced by matrix asymmetry in collocation calculations of vibrational spectra by exploiting near symmetry

Collocation is an enticing alternative to variational methods for solving the vibrational Schrödinger equation. It makes it possible to use a general potential without requiring integrals and quadrature. An important disadvantage of collocation is the need to work with nonsymmetric matrices. Eigenvalues of a large matrix are best computed with an iterative method, but iterative eigensolvers are much more efficient for symmetric matrices. Heretofore, it has been costly to use collocation when the basis set and Hamiltonian matrix are large. We demonstrate that it is possible to systematically make the collocation matrix whose eigenvalues one must compute more and more symmetric and propose an efficient iterative eigensolver for a nearly symmetric matrix. Little is known about exploiting near symmetry. We use a combination of filter diagonalization and an iterative linear solver powered by a three-term recursion relation. We test the ideas with a 6-D Hamiltonian and show that accurate energies are obtained despite the asymmetry.