Methods of solving Coulombic three-body problems in hyperspherical coordinates

A method for solving three-body problems in mass-weighted hyperspherical coordinates in the adiabatic approximation is presented. The adiabatic channel function is expanded in terms of analytical functions expressed in different sets of Jacobi coordinates to describe each dissociation limit naturally. Evaluation of matrix elements between functions in different Jacobi coordinates is achieved through the known transformation properties of hyperspherical harmonics in these coordinates. The method is applied to study the lowest channels for various Coulombic three-body systems such as ${\mathrm{H}}^{\mathrm{-}}$, $e^{\mathrm{-}}$ $e^{\mathrm{-}}$ $e^{+}$, $p^{+}$ $p^{+}$ ${\mu }^{\mathrm{-}}$, $d^{+}$ $d^{+}$ ${\mu }^{\mathrm{-}}$, $e^{+}$ $p^{+}$ $e^{\mathrm{-}}$, $p^{+}$ $d^{+}$ ${\mu }^{\mathrm{-}}$, and $d^{+}$ $t^{+}$ ${\mu }^{\mathrm{-}}$ using only one or two analytical basis functions, and the results are compared with some known calculations. The behavior of the potential curves with respect to the variation of the masses of the three-body system is also examined.