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The functions ϕ$_{a,j}$ ( ω ) are the adiabatic hyperspherical states obtained by solving the three- body Schr¨ odinger equation at fixed ρ = ρ$_{j}$ , i.e. they are eigenstates of H$_{ad}$ at fixed ρ = ρ$_{j}$ with eigenvalues U$_{a}$ ( ρ$_{j}$ ) 

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The functions U$_{a}$ ( ρ$_{j}$ ) are usually referred to as hyperspherical adiabatic potentials. In- serting the expansion of Eq. (11) into the Schr¨ odinger equation Hψ = Eψ reduces the equation to a generalized eigenvalue problem for coefficients c$_{a,j}$ with eigenvalues E 

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where O$_{a}$ ′ j ′ ,aj are the overlap integrals (in the ω space) between adiabatic states ϕ$_{a}$ ′ ,j ′ and ϕ$_{a,j}$ : 

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The subscripts ρ and ω at kets in the above expressions refer to the integration coordinate of the bracket. 

The system (14) of equations resembles to the system of coupled-channel equations, where non-adiabatic couplings 〈 ϕ$_{a}$ ′ | ∂ ∂ρ | ϕ$_{a}$ 〉 d dρ and 〈 ϕ$_{a}$ ′ | ∂ 2 ∂ρ 2 | ϕ$_{a}$ 〉 are replaced with the overlap matrix elements O$_{a}$ ′ j ′ $_{,aj}$. The use of overlap matrix elements instead of the derivatives of adiabatic states with respect to ρ simplifies significantly the numerical solution of the equation [56, 57], and is the main advantage of the SVD method. 

We restricted the present computations to energies and wave functions of bound states for two H − 3 and D − 3 , with total angular momentum J = 0. The hyperangle θ varies in the interval [0, π $_{2}$] while the full interval of variation for the second hyperangle, φ , is [0,2 π ). However, for three identical particles, the interval along φ can be restricted to [ − π $_{2}$, − π $_{6}$] for wave functions of the A ′ 1 or A ′ 2 irreducible representations ( irreps in the following) of the molecular symmetry group ( D$_{3}$$_{h}$ ), and to the interval [ − π $_{2}$,+ π $_{2}$] for wave functions of the E ′ irrep . Irreps with odd parities, A ′′ $_{1}$, A ′′ $_{2}$, and E ′′ are not allowed for J = 0. This is because the inversion applied to the rotational part of the total wave function is reduced to the rotation of the system about the Y axis by π [62]. The J = 0 rotational states are isotropic, 

