Relativistic two-electron atomic and molecular energies using $LS$ coupling and double groups: role of the triplet contributions to singlet states

The triplet contribution is computed to the 1 and 2 ${}^1{S}_0^{\text{e}}$ states of the He atom, to the $1{}^1{S}_0^{\text{e}}$ state of the Li${}^{+}$ and Be${}^{2+}$ ions, and to the $X{}^1{\mathrm{\Sigma }}_{\text{g}}^{+}$ ground state of the H${}_2$ molecule by extensive use of double-group symmetry (equivalent to $LS$ coupling for the atomic systems) during the course of the variational solution of the no-pair Dirac-Coulomb-Breit wave equation. The no-pair Dirac-Coulomb-Breit energies are converged within a sub-parts-per-billion relative precision using an explicitly correlated Gaussian basis optimized to the non-relativistic energies. The $\alpha$ fine-structure constant dependence of the triplet sector contribution to the variational energy is ${\alpha }^4{E}_{\text{h}}$ at leading order, in agreement with the formal perturbation theory result available from the literature.