A general expression for vibrational Hamiltonians expressed in oblique coordinates

We examine the properties of oblique coordinates. The coordinates, introduced by Zúñiga et al. [J. Phys. B: At., Mol. Opt. Phys. 52, 055101, (2019)], reduce vibrational mode-mixing and enhance the quality of vibrational assignments in quantum mechanical investigations of two-dimensional model Hamiltonians. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block-diagonal matrix where the blocks are distinguished by the total quanta of vibrational excitation. Using techniques for the polar decomposition of matrices, we present a scheme for finding these coordinates for systems of arbitrary dimensions. Several molecular examples are presented that highlight the advantages of these coordinates.