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We consider a system of nonlinear coupled equations involving magnetic Schrodinger operators and general potentials. We provide the criteria for the existence of multiple solutions to these equations. As special cases we get the classical results on existence of infinitely many distinct solutions within Hartree and Hartree-Fock theory of atoms and molecules subject to an external magnetic fields. We also extend recent results within this theory, including Coulomb system with a constant magnetic field, a decreasing magnetic field and a "physically measurable" magnetic field.

We study the standard and extended Kohn-Sham models for quasi-relativistic N-electron Coulomb systems; that is, systems where the kinetic energy of the electrons is given by the quasi-relativistic operator root-alpha(-2)Delta(xn) + alpha(-4) - alpha(-2). For spin-unpolarized systems in the local density approximation, we prove existence of a ground state (or minimizer) provided 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) = 2 alpha(-1)pi(-1).

We prove existence of multiple solutions to the quasirelativistic Choquard equation with a scalar potential. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3695991]

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 root-alpha(-2)Delta(xn) + alpha(-4) - alpha(-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 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. (C) 2011 Elsevier Ltd. All rights reserved.

The Complex Absorbing Potential (CAP) method is widely used to compute resonances in Quantum Chemistry, both for scalar valued and matrix valued Hamiltonians. In the semiclassical limit h -> 0 we consider resonances near the real axis and we establish the CAP method rigorously in an abstract matrix valued setting by proving that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proof is based on pseudodifferential operator theory and microlocal analysis.

In the presence of an external magnetic field, we prove existence of a ground state within the Hartree-Fock theory of atoms and molecules. The ground state exists provided the magnetic field decreases at infinity and the total charge Z of K nuclei exceeds N - 1, where N is the number of electrons. In the opposite direction, no ground state exists if N > ; 2Z + K.