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Numerical projection methods are elaborated for the calculation ofeigenstates of the non-relativistic many-particle Coulomb Hamiltonian withselected rotational and parity quantum numbers employing shifted explicitlycorrelated Gaussian functions, which are, in general, not eigenfunctions of thetotal angular momentum and parity operators. The increased computational costof numerically projecting the basis functions onto the irreduciblerepresentations of the three dimensional rotation-inversion group is the priceto pay for the increased flexibility of the basis functions. This increasedflexibility allowed us to achieve a substantial improvement for the variationalupper bound to the Pauli-allowed ground-state energy of theH$_3^+=\{$p$^+,$p$^+,$p$^+,$e$^-,$e$^-\}$ molecular ion treated as an explicitfive-particle system. We compare our pre-Born-Oppenheimer result for thismolecular ion with rovibrational results including non-adiabatic corrections.

Explicitly correlated Gaussian functions with shifted-center and projection techniques in pre-Born-Oppenheimer calculations