309 research outputs found
The Ground State Energy of Heavy Atoms According to Brown and Ravenhall: Absence of Relativistic Effects in Leading Order
It is shown that the ground state energy of heavy atoms is, to leading order,
given by the non-relativistic Thomas-Fermi energy. The proof is based on the
relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum
electrodynamics yielding energy levels correctly up to order Ry
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 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 , where \al denotes the fine structure
constant. We show that, in the simultaneous limit , \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 . 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
Stability of Relativistic Matter With Magnetic Fields
Stability of matter with Coulomb forces has been proved for non-relativistic
dynamics, including arbitrarily large magnetic fields, and for relativistic
dynamics without magnetic fields. In both cases stability requires that the
fine structure constant alpha be not too large. It was unclear what would
happen for both relativistic dynamics and magnetic fields, or even how to
formulate the problem clearly. We show that the use of the Dirac operator
allows both effects, provided the filled negative energy `sea' is defined
properly. The use of the free Dirac operator to define the negative levels
leads to catastrophe for any alpha, but the use of the Dirac operator with
magnetic field leads to stability.Comment: This is an announcement of the work in cond-mat/9610195 (LaTeX
- …