Mueller's Exchange-Correlation Energy in Density-Matrix-Functional Theory

The increasing interest in the Mueller density-matrix-functional theory has led us to a systematic mathematical investigation of its properties. This functional is similar to the Hartree-Fock functional, but with a modified exchange term in which the square of the density matrix \gamma(X, X') is replaced by the square of \gamma^{1/2}(X, X'). After an extensive introductory discussion of density-matrix-functional theory we show, among other things, that this functional is convex (unlike the HF functional) and that energy minimizing \gamma's have unique densities \rho(x), which is a physically desirable property often absent in HF theory. We show that minimizers exist if N \leq Z, and derive various properties of the minimal energy and the corresponding minimizers. We also give a precise statement about the equation for the orbitals of \gamma, which is more complex than for HF theory. We state some open mathematical questions about the theory together with conjectured solutions.Comment: Latex, 42 pages, 1 figure. Minor error in the proof of Prop. 2 correcte

Scott correction for large atoms and molecules in a self-generated magnetic field

We consider a large neutral molecule with total nuclear charge $Z$ in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that Z\al^2\le \kappa_0 for a sufficiently small $\kappa_0$, where \al denotes the fine structure constant. We show that, in the simultaneous limit $Z\to\infty$, \al\to 0 such that \kappa =Z\al^2 is fixed, the ground state energy of the system is given by a two term expansion $c_1Z^{7/3} + c_2(\kappa) Z^2 + o(Z^2)$. The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion

Stability of atoms and molecules in an ultrarelativistic Thomas-Fermi-Weizsaecker model

We consider the zero mass limit of a relativistic Thomas-Fermi-Weizsaecker model of atoms and molecules. We find bounds for the critical nuclear charges that ensure stability.Comment: 8 pages, LaTe

Dipoles in Graphene Have Infinitely Many Bound States

We show that in graphene charge distributions with non-vanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power

Gradient corrections for semiclassical theories of atoms in strong magnetic fields

This paper is divided into two parts. In the first one the von Weizs\"acker term is introduced to the Magnetic TF theory and the resulting MTFW functional is mathematically analyzed. In particular, it is shown that the von Weizs\"acker term produces the Scott correction up to magnetic fields of order $B \ll Z^2$, in accordance with a result of V. Ivrii on the quantum mechanical ground state energy. The second part is dedicated to gradient corrections for semiclassical theories of atoms restricted to electrons in the lowest Landau band. We consider modifications of the Thomas-Fermi theory for strong magnetic fields (STF), i.e. for $B \ll Z^3$. The main modification consists in replacing the integration over the variables perpendicular to the field by an expansion in angular momentum eigenfunctions in the lowest Landau band. This leads to a functional (DSTF) depending on a sequence of one-dimensional densities. For a one-dimensional Fermi gas the analogue of a Weizs\"acker correction has a negative sign and we discuss the corresponding modification of the DSTF functional.Comment: Latex2e, 36 page

Stability and Instability of Relativistic Electrons in Classical Electro magnetic Fields

The stability of matter composed of electrons and static nuclei is investigated for a relativistic dynamics for the electrons given by a suitably projected Dirac operator and with Coulomb interactions. In addition there is an arbitrary classical magnetic field of finite energy. Despite the previously known facts that ordinary nonrelativistic matter with magnetic fields, or relativistic matter without magnetic fields is already unstable when the fine structure constant, is too large it is noteworthy that the combination of the two is still stable provided the projection onto the positive energy states of the Dirac operator, which defines the electron, is chosen properly. A good choice is to include the magnetic field in the definition. A bad choice, which always leads to instability, is the usual one in which the positive energy states are defined by the free Dirac operator. Both assertions are proved here.Comment: LaTeX fil