Calculation of loosely bound levels for threebody quantum systems using hyperspherical coordinates with a mapping procedure
Abstract
In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the threebody system, (2) the solution of the Schroedinger equation in two steps: determination of adiabatic states for a fixed hyperradius and then solution of a set of coupled hyperradial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer {sup 4}He{sub 3}, isotopomers of the H{sub 3}{sup +} ion, and finally a model threebody problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.
 Authors:
 Department of Physics, University of Central Florida, Orlando, Florida 32816 (United States)
 (France)
 Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex (France)
 Publication Date:
 OSTI Identifier:
 20786706
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 1; Other Information: DOI: 10.1103/PhysRevA.73.012702; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 74 ATOMIC AND MOLECULAR PHYSICS; ATOMS; BOUND STATE; COORDINATES; HELIUM; HYDROGEN IONS 3 PLUS; ISOTOPE EFFECTS; MAPPING; MATHEMATICAL SOLUTIONS; MOLECULES; NUCLEONS; SCHROEDINGER EQUATION; THREEBODY PROBLEM; VIBRATIONAL STATES; WAVE FUNCTIONS
Citation Formats
Kokoouline, Viatcheslav, Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex, and MasnouSeeuws, Francoise. Calculation of loosely bound levels for threebody quantum systems using hyperspherical coordinates with a mapping procedure. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.
Kokoouline, Viatcheslav, Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex, & MasnouSeeuws, Francoise. Calculation of loosely bound levels for threebody quantum systems using hyperspherical coordinates with a mapping procedure. United States. doi:10.1103/PHYSREVA.73.0.
Kokoouline, Viatcheslav, Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex, and MasnouSeeuws, Francoise. Sun .
"Calculation of loosely bound levels for threebody quantum systems using hyperspherical coordinates with a mapping procedure". United States.
doi:10.1103/PHYSREVA.73.0.
@article{osti_20786706,
title = {Calculation of loosely bound levels for threebody quantum systems using hyperspherical coordinates with a mapping procedure},
author = {Kokoouline, Viatcheslav and Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex and MasnouSeeuws, Francoise},
abstractNote = {In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the threebody system, (2) the solution of the Schroedinger equation in two steps: determination of adiabatic states for a fixed hyperradius and then solution of a set of coupled hyperradial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer {sup 4}He{sub 3}, isotopomers of the H{sub 3}{sup +} ion, and finally a model threebody problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.},
doi = {10.1103/PHYSREVA.73.0},
journal = {Physical Review. A},
number = 1,
volume = 73,
place = {United States},
year = {Sun Jan 15 00:00:00 EST 2006},
month = {Sun Jan 15 00:00:00 EST 2006}
}

We discuss the bound states of weakly bound van der Waals trimers within the framework of hyperspherical coordinates. The wave function is expanded in terms of hyperspherical harmonics, which form a complete basis set in the angular variables. The resulting set of coupled secondorder differential equations in the hyperradius is solved exactly. Our method gives a value for the zeropoint energy of H{sup +}{sub 3} which is in excellent agreement with previous calculations. For (H{sub 2}){sub 3} and Ne{sub 3}, however, our results show some discrepancy with earlier work.

Calculation of Energy Eigenvalues and Eigenvectors for Three Body Molecules in Jacobi and Hyperspherical Coordinates
The jacobi coordinates is used to eliminate center of mass motion of three body systems. We write the results in hyperspherical coordinates and expand eigenfunction in a series of orthonormal complete set of Y{sub k{alpha}{sub i}} ({omega}{sub i}) in partition i of jacobi coordinates. The matrix elements of two body interaction potential in hyperspherical harmonic approach are determined exactly using computed analytical form of RaynalRevai coefficients to change the base set of Y{sub k{alpha}{sub i}} ({omega}{sub i}) to other set such as Y{sub k{alpha}{sub i}} ({omega}{sub j}. The generalized Laguerre functions are used to change the second order coupled differentialmore » 
Use of alternative hyperspherical coordinates for threebody systems
Relationships between alternative sets of hyperspherical coordinate systems for the treatment of threebody systems are developed. Transformations of hyperspherical harmonics for S states under a change of intrinsic angles are derived, and applied to harmonic expansions of potential energy surfaces. 
Demonstration of moleculelike normal modes of Coulombic threebody systems in hyperspherical coordinates
The adiabatic channel wave functions of Coulombic threebody systems are investigated in massweighted hyperspherical coordinates. We consider the {ital ABA} Coulombic systems, where two of the particles are identical, and examine the density distribution functions at fixed hyperradii {ital R} for different systems as the mass ratio {lambda}={ital m}{sub {ital A}}/{ital m}{sub {ital B}} varies from the atomic limit ({lambda}{r arrow}0, as in H{sup {minus}}) to the molecular limit ({lambda}{r arrow}{infinity}, as in H{sub 2}{sup +}). The bonding and antibonding as well as the rovibrational characters of the threebody systems are illustrated by these density plots. 
Classification of Coulombic threebody systems in hyperspherical coordinates
Hyperspherical coordinates are used to study properties of Coulombic threebody systems of arbitrary masses. Consider a system {ital ABA}, which consists of two identical particles {ital A} and a third particle {ital B}, each with one unit of charge. We examine the evolution of the approximate quantum numbers that are used for classifying bound and resonance states of the system as the mass ratio {lambda}={ital m}{sub {ital A}}/{ital m}{sub {ital B}} changes from the atomic limit ({lambda}{r arrow}0 as in H{sup {minus}}) to the diatomic molecular limit ({lambda}{much gt}1 as in H{sub 2}{sup +}). It is shown that for statesmore »