4 research outputs found
Symmetry adapted ro-vibrational basis functions for variational nuclear motion calculations: TROVE approach
We present a general, numerically motivated approach to the construction of
symmetry adapted basis functions for solving ro-vibrational Schr\"{o}dinger
equations. The approach is based on the property of the Hamiltonian operator to
commute with the complete set of symmetry operators and hence to reflect the
symmetry of the system. The symmetry adapted ro-vibrational basis set is
constructed numerically by solving a set of reduced vibrational eigenvalue
problems. In order to assign the irreducible representations associated with
these eigenfunctions, their symmetry properties are probed on a grid of
molecular geometries with the corresponding symmetry operations. The
transformation matrices are re-constructed by solving over-determined systems
of linear equations related to the transformation properties of the
corresponding wavefunctions on the grid. Our method is implemented in the
variational approach TROVE and has been successfully applied to a number of
problems covering the most important molecular symmetry groups. Several
examples are used to illustrate the procedure, which can be easily applied to
different types of coordinates, basis sets, and molecular systems
Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries
The eigenfunction approach used for discrete symmetries is deduced from the concept of quantum numbers. We show that the irreducible representations (irreps) associated with the eigenfunctions are indeed a shorthand notation for the set of eigenvalues of the class operators (character table). The need of a canonical chain of groups to establish a complete set of commuting operators is emphasized. This analysis allows us to establish in natural form the connection between the quantum numbers and the eigenfunction method proposed by J.Q. Chen to obtain symmetry adapted functions. We then proceed to present a friendly version of the eigenfunction method to project functions