Hybrid variational–perturbation method for calculating ro-vibrational energy levels of polyatomic molecules

Abstract

A procedure for calculation of rotational–vibrational states of medium-sized molecules is presented. It combines the advantages of variational calculations and perturbation theory. The vibrational problem is solved by diagonalising a Hamiltonian matrix, which is partitioned into two sub-blocks. The first, smaller sub-block includes matrix elements with the largest contribution to the energy levels targeted in the calculations. The second, larger sub-block comprises those basis states which have little effect on these energy levels. Numerical perturbation theory, implemented as a Jacobi rotation, is used to compute the contributions from the matrix elements of the second sub-block. Only the first sub-block needs to be stored in memory and diagonalised. Calculations of the vibrational–rotational energy levels also employ a partitioning of the Hamiltonian matrix into sub-blocks, each of which corresponds either to a single vibrational state or a set of resonating vibrational states, with all associated rotational levels. Physically, this partitioning is efficient when the Coriolis coupling between different vibrational states is small. Numerical perturbation theory is used to include the cross-contributions from different vibrational states. Separate individual sub-blocks are then diagonalised, replacing the diagonalisation of a large Hamiltonian matrix with a number of small matrix diagonalisations. Numerical examples show that the proposed hybrid variational–perturbation method greatly speeds up the variational procedure without significant loss of precision for both vibrational–rotational energy levels and transition intensities. The hybrid scheme can be used for accurate nuclear motion calculations on molecules with up to 15 atoms on currently available computers.

Keywords:

• molecular rotation
• vibration
• variational–perturbation theory
• infrared spectra
• nuclear motion

Additional information

Funding

This work was supported by the ERC under Advanced Investigator Project [grant number 267219].