Solutions to quasi-relativistic multi-configurative Hartree–Fock equations in quantum chemistry

We establish the existence of infinitely many distinct solutions to the multi-configurative Hartree–Fock type equations for N -electron Coulomb systems with quasi-relativistic kinetic energy − α − 2 Δ x n + α − 4 − α − 2 for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove the existence of a ground state. The results are valid under the hypotheses that the total charge Z tot of K nuclei is greater than N − 1 and that Z tot is smaller than a critical charge Z c . The proofs are based on a new application of the Lions–Fang–Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert–Riemann manifold, in combination with density operator techniques.