# Rigorous Mathematical Results on Nonlinear Equations Arising in Quantum Chemistry

**Document Type** Theses, Ph.D

Thesis submitted for the award of PhD, to the School of Mathematical Sciences, Dublin Institute of Technology, June 2013.

#### Abstract

It is well-known that the use of relativistic calculations are necessary if one is to obtain an accurate description of heavy atoms and ions. Moreover, the polarization of the electrons must be taken into account if one aims for enhanced accuracy. In this thesis rigorous mathematical results are established for models of atomic and molecular systems when relativistic e↵ects and polarisation are taken into account. (1) The existence of solutions for the spin-polarized Kohn-Sham model is addressed within the Local Density Approximation (LDA) setting and for a unrestricted openshell model. The existence proof is based on the concentration-compactness method, involving two main variables, namely, the spin-up density and the spin-down density, respectively. In the remaining two parts of the thesis, existence of ground states and excited states for heavy atoms is addressed within the context of two di↵erent electronic structure models. Specifically, N-electron Coulomb systems describing heavy atoms are considered when the kinetic energy of an electron is given by the quasi-relativistic operator p−↵−2" + ↵−4 − ↵−2,, where ↵ is Sommerfeld’s fine structure constant. This operator is a nonlocal, pseudodi↵ erential operator of order one. Results on Kohn-Sham, respectively, Hartree- Fock type models are presented, both giving rise to locally compact variational problems with nonlocal operators: (2) For spin-unpolarized systems within the LDA setting, we prove existence of a ground state (or minimizer) for the standard and extended Kohn-Sham model provided that the total charge Ztot of K nuclei is greater than N − 1 and that Ztot is smaller than a critical charge Zc = 2↵−1⇡−1. The proof is based on the concentration-compactness approach. (3) We establish existence of infinitely many distinct solutions of the so-called multiconfigurative Hartree-Fock type equations. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove 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 Zc. 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.

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