Symmetry-Adapted Ro-vibrational Basis Functions for Variational Nuclear Motion Calculations: TROVE Approach

Abstract

We present a general, numerically motivated approach to the construction of symmetry-adapted basis functions for solving ro-vibrational Schrödinger 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 reconstructed by solving overdetermined systems of linear equations related to the transformation properties of the corresponding wave functions on the grid. Our method is implemented in the variational approach TROVE and has been successfully applied to many 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.

Improved potential energy surface and spectral assignments for ammonia in the near-infrared region

“…The stretching and inversion basis functions are generated using the Numerov-Cooley approach [29][30][31] , while for the asymmetric bending modes harmonic oscillators are used. This basis set is processed through a two-step contraction-symmetrisation technique [32] . The final, contracted vibrational ( J = 0 ) basis set used in the refinement comprised all eigenfunctions of the J = 0 Hamiltonian which correspond to the energies below hc · 20 0 0 0 cm −1 .…”

Symmetry Adaptation of the Rotation-Vibration Theory for Linear Molecules

“…However, as the calculations are directly concerned with the rotation and vibration, we characterize them as ro-vibrational. The use of molecular symmetry has applications in diverse fields, including molecular spectroscopy and the construction of molecular wavefunctions, ligand-field theory, material science, and electronic structure calculations [1,[22][23][24][25]. While the use of a symmetry-adapted basis set has been shown to make calculations of ro-vibrational energies far more efficient by reducing the size of the Hamiltonian matrix blocks to be diagonalized [22], it is not strictly necessary.…”

“…The use of molecular symmetry has applications in diverse fields, including molecular spectroscopy and the construction of molecular wavefunctions, ligand-field theory, material science, and electronic structure calculations [1,[22][23][24][25]. While the use of a symmetry-adapted basis set has been shown to make calculations of ro-vibrational energies far more efficient by reducing the size of the Hamiltonian matrix blocks to be diagonalized [22], it is not strictly necessary. This is in contrast to intensity calculations, which would hardly be practicable without knowledge about the symmetry of the ro-vibrational states, mainly due to the selection rules imposed by the nuclear spin statistics associated with different irreducible representations [17,22].…”

“…While the use of a symmetry-adapted basis set has been shown to make calculations of ro-vibrational energies far more efficient by reducing the size of the Hamiltonian matrix blocks to be diagonalized [22], it is not strictly necessary. This is in contrast to intensity calculations, which would hardly be practicable without knowledge about the symmetry of the ro-vibrational states, mainly due to the selection rules imposed by the nuclear spin statistics associated with different irreducible representations [17,22]. A symmetry-free intensity calculation would involve the actual numerical computation of a colossal number of vanishing intensities for transitions that do not satisfy the symmetry selection rules.…”

“…The symmetrization is achieved by utilizing the fact that each of these Hamiltonian operators commutes with the operations in the symmetry group of the molecule in question [22], so that eigenfunctions of a reduced vibrational Hamiltonian generate irreducible representations of the symmetry group. Consequently, we obtain a symmetry-adapted ro-vibrational basis set numerically by solving the eigenvalue problems for the reduced Hamiltonians; the vibrational basis functions are products of the eigenfunctions thus obtained.…”