content_image/373309.jpg content_image/373311.jpg content_image/373314.jpg content_image/373315.jpg content_image/373316.jpg content_image/373317.jpg content_image/373318.jpg
The eigenstates Ψ of three particles interacting through a potential V depending only on the three internuclear distances are obtained by solving the Schr¨ odinger equation with the following Hamiltonian expressed in hyperspherical coordinates ρ, ω in the center-of-mass frame [54]: 2 

where µ = $_{m/}$√$_{3 is the three-body reduced mass and}$ m is the mass of each of the three identical particles. The operator Λ above is the grand angular momentum [58, 59]. It depends only on the set ω of five angles, which include the three Euler angles (for the orientation of the molecular frame in the lab frame) and the hyperangles θ and φ . If the total angular momentum J = 0, Λ depends only on the two hyperangles. The explicit form of Λ 2 is given, for example, by Eq. (27) of Ref. [59]: 

<img src='content_image/373309.jpg'>

<img src='content_image/373311.jpg'>

where J$_{X}$ , J$_{Y}$ , and J$_{Z}$ are the components of the angular momentum along the principal axes of inertia. The orientation of the axes is approximately indicated in Fig. 1. After rescaling the wave function Ψ as Ψ = ρ − 5 / $^{2}$ψ , the Hamiltonian for the new function ψ is written as 

<img src='content_image/373314.jpg'>

<img src='content_image/373315.jpg'>

<img src='content_image/373316.jpg'>

The eigenfunctions ψ ( ρ, ω ) of the above Hamiltonian are sought as an expansion over the basis functions y$_{a,j}$ ( ρ, ω ) with unknown coefficients c$_{a,j}$ : 

The basis functions y$_{a,j}$ ( ρ, ω ) are constructed as products 

<img src='content_image/373317.jpg'>

<img src='content_image/373318.jpg'>

where the functions π$_{j}$ ( ρ ) are DVR-like functions localized at DVR grid points ρ$_{j}$ along the hyper-radius. As in our earlier study [60, 61], here we used the plane wave DVR functions. 

