We report on a series of rigorous results on the existence of ground states and excited states for various weakly coupled, semilinear nonlinear elliptic PDEs arising in electronic structure models of molecular systems in quantum chemistry. 

For wave function methods, we give results for Hartree-Fock type models taking into account relativistic effects and magnetic fields by using the Lions-Fang-Ghoussoub critical point approach to multiple solutions on a noncompact Riemannian manifold. 

Within Density Functional Theory (DFT), we give rigorous results on the open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic $N$-electron Coulomb systems, that is, systems where the kinetic energy of the electrons is given by either the non-relativistic operator $-/Delta_{x_{n}}$ or the quasi-relativistic operator $/sqrt{ -/alpha^{-2} /Delta_{x_{n}} + /alpha^{-4}} -/alpha^{-2}$ 
(nonlocal, pseudodifferential operator of order one); here $/alpha$ is Sommerfeld's fine structure constant. For standard and extended Kohn-Sham models in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge $Z_{/rm tot}$ of $K$ nuclei is greater than $N-1$. For the quasi-relativistic setting we also need that $Z_{/rm tot}$ is smaller than a critical charge $Z_{/rm c}=2 /alpha^{-1} /pi^{-1}$. 

This is joint work with C. Argaez (University of Iceland, Iceland), E. Chiumiento (IAM CONICET, Argentina) and M. Enstedt (Link/"{o}ping University, Sweden). 

References: 

1. C. Argaez, M. Melgaard, /textsl{Existence of a minimizer for the quasi-relativistic Kohn-Sham model}, Electronic  Journal of Differential  Equations Vol. 2012 (2012), 1--20.

2. C. Argaez, M. Melgaard, /textsl{Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry}, Nonlinear Analysis: theory, methods and applications /textbf{75} (2012),  384--404.

3. E. Chiumiento, M. Melgaard, /textsl{Stiefel and Grassmann manifolds in Quantum Chemistry}, J. Geom. Phys. /textbf{62} (2012), no. 8, 1866--1881.

4. M. Enstedt, M. Melgaard, /textsl{Abstract criteria for multiple solutions to nonlinear coupled equations involving  magnetic Schr/"{o}dinger operators},
J. Differential Equations /textbf{253} (2012), 1729--1743. 

5. M. Enstedt, M.Melgaard, /textsl{Existence of infinitely many distinct solutions to the quasi relativistic Hartree-Fock equations}, International Journal of Mathematics and Mathematical Sciences, 2009 (2009), Article ID 651871, 20 pages.

6. M.Enstedt, M.Melgaard, /textsl{Non-existence of a minimizer to the magnetic Hartree-Fock functional}, Positivity /textbf{12} (2008), no 4, 653-666. 

7. M.Enstedt, M.Melgaard, /textsl{Existence of solution to Hartree-Fock equations with decreasing magnetic fields}, Nonlinear Analysis /textbf{69} (2008), no 7, 2125-2141.