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 $\sqrt{ -\a^{-2} \D_{x_{n}} + \a^{-4? }} -\a^{-2}$ for the nth 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 Ztot of K nuclei is greater than N-1 and that Ztot is smaller than a critical charge Ztot. 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.