Algebraic discrete variable representation approaches: application to interatomic effective potentials

Abstract

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.

GRAPHICAL ABSTRACT

Keywords:

• Algebraic approach
• interatomic effective potentials
• discrete variable representation

Acknowledgments

Authors want to thank Octavio Castaños for useful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

Notes: The convergence criterion consisted in obtaining differences $<5\mathrm{\%}$ between theoretical and calculated values. The column ${\overline{E}}_T(v)$ corresponds to the exact values, while ${\overline{E}}_C(v)$ refers to the calculated energies provided by the different algebraic DVR methods. For the $SU(2)$-UGA method N stands for the total number of bosons, while in the other two cases, N + 1 corresponds to the dimension of the basis.

Notes: The convergence criterion consisted in obtaining errors $<5\mathrm{\%}$ between theoretical and calculated value. The column ${\overline{E}}_T(v)$ corresponds to the exact values, while ${\overline{E}}_C(v)$ refers to the calculated energies provided by the different algebraic DVR methods.

Notes: The convergence criterion consisted in obtaining errors $<5\mathrm{\%}$ between theoretical and calculated value. The column ${\overline{E}}_T(v)$ corresponds to the exact values, while ${\overline{E}}_C(v)$ refers to the calculated energies provided by the different algebraic DVR methods.

Notes: Since analytical energies are not available, the convergence criterion is based on the coincidence of the energy levels for the first five states. ${\overline{E}}_C(v)$ refers to the calculated energies provided by the different algebraic DVR methods.

Notes: The first column indicates the quantum number together with the parity. ${\overline{E}}_T(v)$ refers to the exact energy while ${\overline{E}}_C(v)$ stands for calculated energies provided by the different algebraic DVR methods. The first seven states are well described with relatively low basis dimension, but the last two states needs a larger number of basis functions, which is indicated by the additional row including the basis dimension.

Notes: The dimension of the bases was determined by the criterion of describing the wave function for the last state in each case. ${\overline{E}}_T(v)$ refers to the exact energy when it exists, while ${\overline{E}}_C(v)$ stands for calculated energies provided by the different algebraic DVR methods.

Notes: The convergence criterion consisted in obtaining differences $<5\mathrm{\%}$ between theoretical and calculated values for L = 0. The column ${\overline{E}}_T(v,0)$ corresponds to the exact values with L = 0, while ${\overline{E}}_C(v,0)$ refers to the calculated energies provided by the different algebraic DVR methods. For the $SU(4)$-UGA method stands for the total number of bosons N, while in the HO-DVR method corresponds to the basis dimension.

Note: For the $SU(4)$-UGA method the total number of bosons was taken to be N = 391, while in the HO-DVR case the basis dimension was N = 106.

Notes: The first column indicates the quantum number together with the parity associated with the 1D case. ${\overline{E}}_T(v)$ refers to the exact energy for L = 0, while ${\overline{E}}_C(v)$ stands for calculated energies.

Notes: The column ${\overline{E}}_T(v,0)$ corresponds to the exact values with L = 0, while ${\overline{E}}_C(v,0)$ refers to the calculated energies provided by the different algebraic DVR methods. For the $SU(4)$-UGA method stands for the total number of bosons N, while in the other case corresponds to the dimension of the basis.

Notes: For the $SU(4)$-UGA method the total number of bosons N = 71 was taken, while in the HO-DVR case the dimension of the basis was taken to be N = 38.

Notes: We present the calculation with the $SU(4)$-UGA and the HO-DVR method. We increased the total number of bosons and the basis dimension in order to observe convergence in the energy.

Notes: The column ${\overline{E}}_T(v,0)$ corresponds to the exact values with L = 0. For the $SU(4)$-UGA method the total number of bosons N = 2501 was taken, while in the HO-DVR case the dimension of the basis was taken N = 100.

Additional information

Funding

This work is partially supported by DGAPA-UNAM, Mexico (Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México), under project IN-212020, DGAPA UNAM for postdoctoral scholarship (Facultad de Química) to second author, by Spanish Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía [grant numbers FQM-160], and by the Spanish Ministerio de Ciencia e Innovación, ref. FIS2017-88410-P, PID2019-104002GB-C22, and FPA2016-77689-C2-1-R and by the European Commission, ref. H2020-INFRAIA-2014-2015 (ENSAR2). First author is also grateful for the scholarship (Posgrado en Ciencia e Ingeniería de Materiales) provided by CONACyT, México.