The simplest, algebraic quantum-electrodynamical corrections due to the double-negative energy subspace and instantaneous interactions are computed to the no-pair energy of two-spin-1/2-fermion systems. Numerical results are reported for two-electron atoms with a clamped nucleus and positronium-like genuine two-particle systems. The Bethe-Salpeter equation provides the theoretical framework, and numerical methods have been developed for its equal-time time slice. In practice, it requires solving a 16-component eigenvalue equation with a two-particle Dirac Hamiltonian, including the appropriate interaction. The double-pair corrections can either be included in the interaction part of the eigenvalue equation or treated as a perturbation to the no-pair Hamiltonian. The numerical results have an $\alpha$ fine-structure constant dependence that is in excellent agreement with the known ${\alpha }^3{E}_{\mathrm{h}}$-order double-pair correction of nonrelativistic quantum electrodynamics.

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• Dirac equation
• Electronic structure of atoms & molecules
• Quantum electrodynamics