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Variational and perturbative relativistic energies are computed and comparedfor two-electron atoms and molecules with low nuclear charge numbers. Ingeneral, good agreement of the two approaches is observed. Remaining deviationscan be attributed to higher-order relativistic, also called non-radiativequantum electrodynamics (QED), corrections of the perturbative approach thatare automatically included in the variational solution of the no-pairDirac$-$Coulomb$-$Breit (DCB) equation to all orders of the $\alpha$fine-structure constant. The analysis of the polynomial $\alpha$ dependence ofthe DCB energy makes it possible to determine the leading-order relativisticcorrection to the non-relativistic energy to high precision withoutregularization. Contributions from the Breit$-$Pauli Hamiltonian, for whichexpectation values converge slowly due the singular terms, are implicitlyincluded in the variational procedure. The $\alpha$ dependence of the no-pairDCB energy shows that the higher-order ($\alpha^4 E_\mathrm{h}$) non-radiativeQED correction is 5 % of the leading-order ($\alpha^3 E_\mathrm{h}$)non-radiative QED correction for $Z=2$ (He), but it is 40 % already for $Z=4$(Be$^{2+}$), which indicates that resummation provided by the variationalprocedure is important already for intermediate nuclear charge numbers.

Variational versus perturbative relativistic energies for small and light atomic and molecular systems