A new method for solving an eigenvalue problem for a system of three Coulomb particles within the hyperspherical adiabatic representation

The quantum mechanical three-body problem with Coulomb interaction is formu lated within the adiabatic representation method using the hyperspherical c oordinates. The Kantorovich method of reducing the multidimensional problem to the one-dimensional one is used. A new method for computing variable co efficients (potential matrix elements of radial coupling) of a resulting sy stem of ordinary second-order differential equations is proposed. It allows the calculation of the coefficients with the same precision as the adiabat ic functions obtained as solutions of an auxiliary parametric eigenvalue pr oblem. In the method proposed, a new boundary parametric problem with respe ct to unknown derivatives of eigensolutions in the adiabatic variable (hype rradius) is formulated. An efficient, fast, and stable algorithm for solvin g the boundary problem with the same accuracy for the adiabatic eigenfuncti ons and their derivatives is proposed. The method developed is tested on a parametric eigenvalue problem for a hydrogen atom on a three-dimensional sp here that has an analytical solution. The accuracy, efficiency, and robustn ess of the algorithm are studied in detail. The method is also applied to t he computation of the ground-state energy of the helium atom and negative h ydrogen ion. (C) 2000 Academic Press.