Periodic Hartree-Fock theory

The most commonly used approximations to the electronic ground state energy of a quantum mechanical model can schematically be classified into two main classes: - Density functional methods are used to investigate the electronic structure of many-body systems. - Wave function methods aim at finding an approximation of the ground state wave function and the ground state energy of a quantum many-body system. The conventional wave function approaches use wave function as the central quantity, since it contains the full information of a system. They assure that the energy computed from a guessed wave function is an upper bound to the true ground state energy. Full minimization of the energy with respect to all allowed wave functions will give the true ground state. The Hartree-Fock approximation, introduced by Hartree and improved by Fock in the late 1920s, is an important example of these methods. We are interested in the second method, especially in the periodic Hartree-Fock theory. The present thesis includes three parts, the periodic minimizer of the Hartree-Fock (HF) functional and its properties, the Bloch wave decomposition and its application on the periodic HF theory and the opening gap in the spectrum of the fibered HF Hamiltonian in the presence of a weak, periodic one-dimensional potential. The overview is organized as follows. In Chapter 1 an introduction to HF theory and its importance as a wave function method for finding an approximation of the electronic ground state energy is provided. Here the concepts of the ground state energy and the HF energy of a Hamiltonian acting on the fermion Fock space are presented. For the readers convenience some results are recalled without proofs due to Lieb and Solovej concerning the existence of the HF minimizer. The rest of Chapter 1 focuses on the periodic HF theory, where the periodic model and the corresponding variational problem are introduced. Chapter 2 is devoted to the study of the properties of the periodic HF minimizer. The existence of the periodic HF minimizer using arguments similar to those of Catto, Le Bris and Lions is achieved. The basic strategy in this proof is to consider a net {γn}nelementofI2 of density matrices in the variational set of periodic minimizers Pper(N) such that the HF functional εhf(γn) tends to the periodic ground state energy Ehfper(N) as n tends to infinity. Then it is proved that this net converges, up to the extraction of a subnet, to some operators γelement of Pper(N) satisfying εhf(γ)=Ehfper(N). The latter can be seen by constructing a weak* topology on Pper(N) and applying Arazy's theorem to obtain strong convergence. Moreover, the fact that the periodic minimizer is a projection onto the N lowest eigenvalues of the periodic HF Hamiltonian is verified. The presented proof is an adaptation of the one given by Bach, Froehlich and Jonsson, restricted to the periodic case. Furthermore it is proven by contradiction to the minimality of the periodic minimizer that there is a gap in the spectrum of the periodic HF Hamiltonian above the N-th energy level. In addition, the uniqueness of the minimizer on Pper(N) is shown by using the self-consistent equation it satisfies and the contraction mapping principle as in the work of Griesemer and Hantsch, which was based on the paper of Huber and Siedentop on solutions of the Dirac-Fock equations. Here the assumption that the N-th eigenvalue of h is separated by a gap of a positive size from the rest of the spectrum is essential. The presence of such a gap implies that the energy increases by moving from the periodic minimizer even in the set of non-periodic matrices, which means that the HF and the periodic HF functional coincide at the periodic minimizer. In Chapter 3 the periodic properties of the HF minimizer on hΛ=L2(Λ) for a given torus Λ=(R/(LZ)d) is studied. A unit cube Q=Λ/Γ and a lattice Γ=(qZ)d/(LZ)d of Λ are introduced to decompose hΛ according to the translational invariance by vectors of Γ. After this decomposition of functions in hΛ into Bloch waves a direct integral decomposition of operators K on hΛ can be derived, in the sense that the spectral analysis of K reduces to the spectral analysis of its fibers. Applying this construction to the periodic density matrices yields an equivalent statement for their periodicity. Moreover, a version of Bloch's theorem adapted to our framework is given. It states that every eigenvector of a Hamiltonian with periodic potential can be chosen in the form of a wave function, which is a multiplication of a function having the same periodicity as the potential with the complex phase of a plane wave of absolute value one (Bloch's theorem). Chapter 4 is dedicated to the characterization of the periodic HF minimizers. The fibers of the periodic HF Hamiltonian and that of the periodic HF functional are explicitly computed. Theses expressions are used to generalize Lieb's variational principle in the periodic case. By using this proof, an estimate on the distance between the N+1-th and the N-th eigenvalue of the fibered Hamiltonian is obtained in terms of the corresponding fibered periodic potential. In the last chapter another model is studied. The Hilbert space of states is given by hΛ=l2(Λ) where Λ=Zd/(LZ)d is a discrete torus and the Hamiltonian consists of the discrete Laplace operator plus an interaction which is identified with a multiplication operator with a positive symmetric function W:Λ → R+. (orig.)