_{3}using hyperspherical coordinates and a slow variable discretization approach

^{1,a)}

### Abstract

We study theoretically the ground and excited bound states of the bosonic rare gas van der Waals trimer Ne_{3}. A slow variable discretization approach is adopted to solve the nuclear Schrödinger equation, in which the Schrödinger equation in hyperangular coordinates is solved using basis splines at a series of fixed finite-element methods discrete variable representation hyper-radii. We consider not only zero total nuclear orbital angular momentum,*J* = 0, states but also *J* > 0 states. By using the best empirical neon dimer interaction potentials, all the bound state energy levels of Ne_{3} will be calculated for total angular momenta up to *J* = 6, as well as their average root-mean-square radii. We also analyze the wave functions in hyperspherical coordinates for several selected bound states.

Part of the computations were performed using Research Center for Computational Science, Okazaki, Japan.

I. INTRODUCTION

II. METHOD

III. RESULTS

A. Bound state energy levels

B. Total wave functions and molecular structures

IV. SUMMARY

### Key Topics

- Bound states
- 46.0
- Wave functions
- 12.0
- Angular momentum
- 6.0
- Finite element methods
- 6.0
- Coupled channel methods
- 5.0

## Figures

Two-dimensional contour plot of the potential surface at a fixed hyper-radius *R* = 10*a* _{0} as a function of the hyperangles θ and φ for the Ne_{3} system. The dotted line indicates the lowest contour line (−50 cm^{−1}) and the dashed line the second lowest contour line (−40 cm^{−1}). The other contour lines correspond to −30, −20, and −10 cm^{−1}.

Two-dimensional contour plot of the potential surface at a fixed hyper-radius *R* = 10*a* _{0} as a function of the hyperangles θ and φ for the Ne_{3} system. The dotted line indicates the lowest contour line (−50 cm^{−1}) and the dashed line the second lowest contour line (−40 cm^{−1}). The other contour lines correspond to −30, −20, and −10 cm^{−1}.

Adiabatic hyperspherical potential curves *U* _{ν}(*R*) of the Ne_{3} system for (a) , (b) , (c) , and (d) . In each figure, the 40 lowest potential curves (ν = 0 − 39) are shown.

Adiabatic hyperspherical potential curves *U* _{ν}(*R*) of the Ne_{3} system for (a) , (b) , (c) , and (d) . In each figure, the 40 lowest potential curves (ν = 0 − 39) are shown.

Bound state energy levels of Ne_{3} as functions of the total nuclear orbital angular momentum *J* and the parity Π.

Bound state energy levels of Ne_{3} as functions of the total nuclear orbital angular momentum *J* and the parity Π.

Contour plots of the 2D probability density functions for the *n* = 0 and *n* = 10 bound states of Ne_{3}() (a) in *R*θ-space, (b) in θφ-space, and (c) in *R*φ-space. The *n* = 0 state is indicated by solid lines, the *n* = 10 state by dashed lines.

Contour plots of the 2D probability density functions for the *n* = 0 and *n* = 10 bound states of Ne_{3}() (a) in *R*θ-space, (b) in θφ-space, and (c) in *R*φ-space. The *n* = 0 state is indicated by solid lines, the *n* = 10 state by dashed lines.

Contour plots of the 2D probability density functions for the *n* = 0 and *n* = 10 bound states of Ne_{3}() (a) in *R*θ-space, (b) in θφ-space, and (c) in *R*φ-space. The *n* = 0 state is indicated by solid lines, the *n* = 10 state by dashed lines.

*n* = 0 and *n* = 10 bound states of Ne_{3}() (a) in *R*θ-space, (b) in θφ-space, and (c) in *R*φ-space. The *n* = 0 state is indicated by solid lines, the *n* = 10 state by dashed lines.

Contour plots of the 2D probability density functions for the *n* = 0, *n* = 10, and *n* = 20 bound states of Ne_{3}() (a) in *R*θ-space, (b) in θφ-space, and (c) in *R*φ-space. The *n* = 0 state is indicated by solid lines, the *n* = 10 state by dashed lines, and the *n* = 20 state by dashed-dotted lines.

Contour plots of the 2D probability density functions for the *n* = 0, *n* = 10, and *n* = 20 bound states of Ne_{3}() (a) in *R*θ-space, (b) in θφ-space, and (c) in *R*φ-space. The *n* = 0 state is indicated by solid lines, the *n* = 10 state by dashed lines, and the *n* = 20 state by dashed-dotted lines.

## Tables

Effects of permutation operations on the hyperangles φ, α, β, γ and the Wigner *D* function.

Effects of permutation operations on the hyperangles φ, α, β, γ and the Wigner *D* function.

Energy levels of selected bound states of Ne_{3} at various levels of approximation. The energies are given in units of cm^{−1} and are relative to the three-body dissociation threshold.

Energy levels of selected bound states of Ne_{3} at various levels of approximation. The energies are given in units of cm^{−1} and are relative to the three-body dissociation threshold.

Bound state energies of Ne_{3} for and 1^{±}. The energies are given in units of cm^{−1}, and are relative to the three-body dissociation limit. The results using the Morse potential are also included. The present results are compared with those based on the FEM (Ref. 18), DGF (Ref. 17), and Pekeris coordinate (Ref. 4) approaches.

Bound state energies of Ne_{3} for and 1^{±}. The energies are given in units of cm^{−1}, and are relative to the three-body dissociation limit. The results using the Morse potential are also included. The present results are compared with those based on the FEM (Ref. 18), DGF (Ref. 17), and Pekeris coordinate (Ref. 4) approaches.

Bound state energies of Ne_{3} for and 3^{±}. The energies are given in units of cm^{−1}, and are relative to the three-body dissociation limit. The present results are compared with those based on the FEM (Ref. 18).

Bound state energies of Ne_{3} for and 3^{±}. The energies are given in units of cm^{−1}, and are relative to the three-body dissociation limit. The present results are compared with those based on the FEM (Ref. 18).

Bound state energies of Ne_{3} for , 5^{±}, and 6^{±}. The energies are given in units of cm^{−1}, and are relative to the three-body dissociation limit.

Bound state energies of Ne_{3} for , 5^{±}, and 6^{±}. The energies are given in units of cm^{−1}, and are relative to the three-body dissociation limit.

Average root-mean-square radii of Ne_{3} for , 1^{±}, and 2^{±}. The root-mean-square radii are given in units of Bohr radius *a* _{0}. The results using the Morse potential are also included. The present results are compared with those based on the FEM (Ref. 18), DGF (Ref. 17), and Pekeris coordinate (Ref. 4) approaches.

Average root-mean-square radii of Ne_{3} for , 1^{±}, and 2^{±}. The root-mean-square radii are given in units of Bohr radius *a* _{0}. The results using the Morse potential are also included. The present results are compared with those based on the FEM (Ref. 18), DGF (Ref. 17), and Pekeris coordinate (Ref. 4) approaches.

Average root-mean-square radii of Ne_{3} for . The root-mean-square radii are given in units of Bohr radius *a* _{0}.

Average root-mean-square radii of Ne_{3} for . The root-mean-square radii are given in units of Bohr radius *a* _{0}.

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