Algebraic discrete variable representation approaches: application to interatomic effective potentials

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Algebraic approaches based on a discrete variable representation scheme (DVR) are proposed to describe interatomic effective potentials. The recently proposed unitary group approach (UGA) is identified as a discrete variable representation method among other more general methods discussed in this work. The basic idea consists in establishing an algebraic realisation of the coordinate and momentum in the framework of a dynamical group associated with a complete basis. In the context of the algebraic DVR methods, the discrete coordinate and momentum representations are obtained by diagonalising their matrix representation in a basis associated with a specific potential, in particular, harmonic oscillator, Morse and Pöschl-Teller potentials. This feature provides useful tools to obtain the matrix representation of the Hamiltonian in a simple form through the use of the transformation coefficients connecting the different bases. The proposed algebraic DVR approaches are applied and compared in the framework of 1,2 and 3-D systems. The different approaches are evaluated by considering the Morse, Pöschl-Teller, Kratzer-Fues, Deng-Fan, Varshni and Coulomb potentials.