Using a procedure originally due to Hylleraas, a convenient expansion
in coupled spherical harmonics which terminates in a very small number
of terms is applied to the treatment of fully correlated two-electron
wave functions of any symmetry (total angular momentum, parity, and s
pin). Coupled equations satisfied by these wave functions are derived
which are well adapted to computation and which we discretize on a num
erical lattice utilizing the basis-spline collocation method. Use of t
his method which relies on very flexible basis functions is intended t
o facilitate subsequently the consideration of time-dependent rearrang
ements such as autoionization, photoionization, and electron-impact ex
citation and ionization. Here, we describe the underlying theoretical
and computational methods concerning our treatment of the two-electron
problem, the lattice discretization, and partial eigensolution by dam
ped relaxation. Results of explicit calculations are given regarding t
he ground state and two low-lying singly excited states of helium.