  We give a detailed account of an $\it{ab}$ $\it{initio}$ spectral approach
for the calculation of energy spectra of two active electron atoms in a system
of hyperspherical coordinates. In this system of coordinates, the Hamiltonian
has the same structure as the one of atomic hydrogen with the Coulomb potential
expressed in terms of a hyperradius and the nuclear charge replaced by an angle
dependent effective charge. The simplest spectral approach consists in
expanding the hyperangular wave function in a basis of hyperspherical
harmonics. This expansion however, is known to be very slowly converging.
Instead, we introduce new hyperangular sturmian functions. These functions do
not have an analytical expression but they treat the first term of the
multipole expansion of the electron-electron interaction potential, namely the
radial electron correlation, exactly. The properties of these new functions are
discussed in detail. For the basis functions of the hyperradius, several
choices are possible. In the present case, we use Coulomb sturmian functions of
half integer angular momentum. We show that, in the case of H$^-$, the accuracy
of the energy and the width of the resonance states obtained through a single
diagonalization of the Hamiltonian, is comparable to the values given by
state-of-the-art methods while using a much smaller basis set. In addition, we
show that precise values of the electric-dipole oscillator strengths for
$S\rightarrow P$ transitions in helium are obtained thereby confirming the
accuracy of the bound state wave functions generated with the present method.
