  Numerical projection methods are elaborated for the calculation of
eigenstates of the non-relativistic many-particle Coulomb Hamiltonian with
selected rotational and parity quantum numbers employing shifted explicitly
correlated Gaussian functions, which are, in general, not eigenfunctions of the
total angular momentum and parity operators. The increased computational cost
of numerically projecting the basis functions onto the irreducible
representations of the three dimensional rotation-inversion group is the price
to pay for the increased flexibility of the basis functions. This increased
flexibility allowed us to achieve a substantial improvement for the variational
upper bound to the Pauli-allowed ground-state energy of the
H$_3^+=\{$p$^+,$p$^+,$p$^+,$e$^-,$e$^-\}$ molecular ion treated as an explicit
five-particle system. We compare our pre-Born-Oppenheimer result for this
molecular ion with rovibrational results including non-adiabatic corrections.
