32 research outputs found
Explicit large nuclear charge limit of electronic ground states for Li, Be, B, C, N, O, F, Ne and basic aspects of the periodic table
This paper is concerned with the Schrödinger equation for atoms and ions with to 10 electrons. In the asymptotic limit of large nuclear charge , we determine explicitly the low-lying energy levels and eigenstates. The asymptotic energies and wavefunctions are in good quantitative agreement with experimental data for positive ions, and in excellent qualitative agreement even for neutral atoms (). In particular, the predicted ground state spin and angular momentum quantum numbers ( for He, Be, Ne, for H and Li, for N, for B and F, and for C and O) agree with experiment in every case. The asymptotic Schrödinger ground states agree, up to small corrections, with the semiempirical hydrogen orbital configurations developed by Bohr, Hund, and Slater to explain the periodic table. In rare cases where our results deviate from this picture, such as the ordering of the lowest and states of the carbon isoelectronic sequence, experiment confirms our predictions and not Hund's
Radio emission of sea surface at centimeter wavelengths and is fluctuations
The eigen thermal radio emission of the sea was examined as well as the agitated surface of the sea when the reflection (scattering) is similar in nature to diffused scattering. The contribution of this emission to the total emission of the sea is practically constant in time, and the time fluctuations of the radio emissions of the sea are basically determined only by a change in the eigen emission of the sea, connected with the agitation
On the Essential Spectrum of Many-Particle Pseudorelativistic Hamiltonians with Permutational Symmetry Account
In this paper we formulate our results on the essential spectrum of many-particle pseudorelativistic Hamiltonians without magnetic and external potential fields in the spaces of functions, having arbitrary type α of the permutational symmetry. We discover location of the essential spectrum for all α and for some cases we establish new properties of the lower bound of this spectrum, which are useful for study of the discrete spectrum
Bound States at Threshold resulting from Coulomb Repulsion
The eigenvalue absorption for a many-particle Hamiltonian depending on a
parameter is analyzed in the framework of non-relativistic quantum mechanics.
The long-range part of pair potentials is assumed to be pure Coulomb and no
restriction on the particle statistics is imposed. It is proved that if the
lowest dissociation threshold corresponds to the decay into two likewise
non-zero charged clusters then the bound state, which approaches the threshold,
does not spread and eventually becomes the bound state at threshold. The
obtained results have applications in atomic and nuclear physics. In
particular, we prove that atomic ion with atomic critical charge and
electrons has a bound state at threshold given that , whereby the electrons are treated as fermions and the mass of the
nucleus is finite.Comment: This is a combined and updated version of the manuscripts
arXiv:math-ph/0611075v2 and arXiv:math-ph/0610058v
Efficient Algorithm for Asymptotics-Based Configuration-Interaction Methods and Electronic Structure of Transition Metal Atoms
Asymptotics-based configuration-interaction (CI) methods [G. Friesecke and B.
D. Goddard, Multiscale Model. Simul. 7, 1876 (2009)] are a class of CI methods
for atoms which reproduce, at fixed finite subspace dimension, the exact
Schr\"odinger eigenstates in the limit of fixed electron number and large
nuclear charge. Here we develop, implement, and apply to 3d transition metal
atoms an efficient and accurate algorithm for asymptotics-based CI.
Efficiency gains come from exact (symbolic) decomposition of the CI space
into irreducible symmetry subspaces at essentially linear computational cost in
the number of radial subshells with fixed angular momentum, use of reduced
density matrices in order to avoid having to store wavefunctions, and use of
Slater-type orbitals (STO's). The required Coulomb integrals for STO's are
evaluated in closed form, with the help of Hankel matrices, Fourier analysis,
and residue calculus.
Applications to 3d transition metal atoms are in good agreement with
experimental data. In particular we reproduce the anomalous magnetic moment and
orbital filling of Chromium in the otherwise regular series Ca, Sc, Ti, V, Cr.Comment: 14 pages, 1 figur
Heat kernel estimates and spectral properties of a pseudorelativistic operator with magnetic field
Based on the Mehler heat kernel of the Schroedinger operator for a free
electron in a constant magnetic field an estimate for the kernel of E_A is
derived, where E_A represents the kinetic energy of a Dirac electron within the
pseudorelativistic no-pair Brown-Ravenhall model. This estimate is used to
provide the bottom of the essential spectrum for the two-particle
Brown-Ravenhall operator, describing the motion of the electrons in a central
Coulomb field and a constant magnetic field, if the central charge is
restricted to Z below or equal 86
Rigorous conditions for the existence of bound states at the threshold in the two-particle case
In the framework of non-relativistic quantum mechanics and with the help of
the Greens functions formalism we study the behavior of weakly bound states as
they approach the continuum threshold. Through estimating the Green's function
for positive potentials we derive rigorously the upper bound on the wave
function, which helps to control its falloff. In particular, we prove that for
potentials whose repulsive part decays slower than the bound states
approaching the threshold do not spread and eventually become bound states at
the threshold. This means that such systems never reach supersizes, which would
extend far beyond the effective range of attraction. The method presented here
is applicable in the many--body case
Zero Energy Bound States in Many--Particle Systems
It is proved that the eigenvalues in the N--particle system are absorbed at
zero energy threshold, if none of the subsystems has a bound state with and none of the particle pairs has a zero energy resonance. The pair
potentials are allowed to take both signs
Binding of Polarons and Atoms at Threshold
If the polaron coupling constant is large enough, bipolarons or
multi-polarons will form. When passing through the critical from
above, does the radius of the system simply get arbitrarily large or does it
reach a maximum and then explodes? We prove that it is always the latter. We
also prove the analogous statement for the Pekar-Tomasevich (PT) approximation
to the energy, in which case there is a solution to the PT equation at
. Similarly, we show that the same phenomenon occurs for atoms, e.g.,
helium, at the critical value of the nuclear charge. Our proofs rely only on
energy estimates, not on a detailed analysis of the Schr\"odinger equation, and
are very general. They use the fact that the Coulomb repulsion decays like
, while `uncertainty principle' localization energies decay more rapidly,
as .Comment: 19 page
On the Quantum Theory of Molecules
Transition state theory was introduced in the 1930s to account for chemical
reactions. Central to this theory is the idea of a potential energy surface
(PES). It was assumed that such a surface could be constructed using
eigensolutions of the Schr\"{o}dinger equation for the molecular (Coulomb)
Hamiltonian but at that time such calculations were not possible. Nowadays
quantum mechanical ab-initio electronic structure calculations are routine and
from their results PESs can be constructed which are believed to approximate
those assumed derivable from the eigensolutions. It is argued here that this
belief is unfounded. It is suggested that the potential energy surface
construction is more appropriately regarded as a legitimate and effective
modification of quantum mechanics for chemical purpose