Optimized coordinates for anharmonic vibrational structure theories

A procedure to determine optimal vibrational coordinates is developed on the basis of an earlier idea of Thompson and Truhlar [J. Chem. Phys. 77, 3031 (1982)]10.1063/1.444226. For a given molecule, these coordinates are defined as the unitary transform of the normal coordinates that minimizes the energy of the vibrational self-consistent-field (VSCF) method for the ground state. They are justified by the fact that VSCF in these coordinates becomes exact in two limiting cases: harmonic oscillators, where the optimized coordinates are normal, and noninteracting anharmonic oscillators, in which the optimized coordinates are localized on individual oscillators. A robust and general optimization algorithm is developed, which decomposes the transformation matrix into a product of Jacobi matrices, determines the rotation angle of each Jacobi matrix that minimizes the energy, and iterates the process until a minimum in the whole high dimension is reached. It is shown that the optimized coordinates are neither entirely localized nor entirely delocalized (or normal) in any of the molecules (the water, water dimer, and ethylene molecules) examined (apart from the aforementioned limiting cases). Rather, high-frequency stretching modes tend to be localized, whereas low-frequency skeletal vibrations remain normal. On the basis of these coordinates, we introduce two new vibrational structure methods: optimized-coordinate VSCF (oc-VSCF) and optimized-coordinate vibrational configuration interaction (oc-VCI). For the modes that become localized, oc-VSCF is found to outperform VSCF, whereas, for both classes of modes, oc-VCI exhibits much more rapid convergence than VCI with respect to the rank of excitations. We propose a rational configuration selection for oc-VCI when the optimized coordinates are localized. The use of the optimized coordinates in VCI with this configuration selection scheme reduces the mean absolute errors in the frequencies of the fundamentals and the first overtones/combination tones from 104.7 (VCI) to 10.7 (oc-VCI) and from 132.4 (VCI) to 8.2 (oc-VCI) cm^-1 for the water molecule and the water dimer, respectively. It is also shown that the degree of coupling in the potential for ethylene is reduced effectively from four modes to three modes by the transformation from the normal to optimized coordinates, which enhances the accuracy of oc-VCI with low-rank excitations.