Approach to the Bound-State Three-Body Problem with Application to the Helium-Like Atom

A technique is presented for treating a general type of three-body bound-state problem for situations where the interaction may be written as the sum of three pair potentials. The method is based on the work of Eyges and consists of writing the total wave function for the three-body problem in a special form, i.e., as the sum of three different parts or "orbitals" defined in a natural way from the integral equation by using the appropriate symmetrized or antisymmetrized Green's function. A set of three integral equations for the three orbitals is derived: first for the situation where each particle is distinguishable, and then for situations where two or all of the three particles are identical. It is found that, when some of the particles are identical and the Pauli principle is incorporated into the formulation, the number of independent orbitals can be reduced. Some simple one-dimensional applications involving $\delta$-function pair potentials are examined. The helium atom is discussed from a three-body point of view in order to illustrate our general apporach to the above-mentioned set of coupled integral equations. Each orbital is expanded into a complete set of two-body Sturmian functions for the Coulomb potential. For states in atomic helium of the form $L=0\mathrm{}$, the equations assume a simplified form. For these states, the infinite set of integral equations in one variable generated by the expansion is truncated at several orders and solved numerically. Rapid convergence is demonstrated for low-lying states of the helium-like atoms using this method. In particular, results are reported for the $1s1s^{1}S$, $1s2s^{1}S$, and $2s2s^{1}S$ states of He, ${\mathrm{Li}}^{+}$, and ${\mathrm{H}}^{-}$. The technique is not based upon a perturbation expansion or a variational principle.