ABSTRACT
A general approach to describe the vibrational excitations of semirigid molecules is presented. The method is based on a local mode description of vibrations in terms of an algebraic representation of Morse and/or Pöschl-Teller potentials. Expansions in terms of internal coordinates of the kinetic and potential energies of the vibrational Hamiltonian are considered up to a given order (usually quartic order). The local Morse coordinates https://www.w3.org/1998/Math/MathML"> y i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq974.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> as well as the momenta https://www.w3.org/1998/Math/MathML"> p i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq975.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are thereafter expanded in terms of creation and annihilation operators of the Morse functions keeping terms up to order https://www.w3.org/1998/Math/MathML"> 1 / κ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq976.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (up to quadratic terms in the operators), where κ is the Child's parameter related with the depth of the potential. The matrix elements are calculated taking advantage of https://www.w3.org/1998/Math/MathML"> su ( 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq977.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> dynamical symmetry of both potentials. In this framework, terms conserving the polyad pseudo quantum number can be considered. Unlike the description in terms of a harmonic basis, all the force constants up to quartic order can be determined. In addition, a tensorial formalism to expand the Hamiltonian is presented. This approach allows to identify higher order interactions involved in the algebraic expansions of the coordinates and momenta. Group theoretical techniques are used to establish the symmetry adapted basis as well as the interactions. A basis carrying normal and local labels is constructed. This approach allows to establish a method to eliminate the spurious modes both from the Hamiltonian and the basis.
