626 research outputs found
PERTURBATION THEORY FOR ILLUSTRATION PURPOSES
Perturbation theory applied to laguerre function
Hadronic Effects in the Pionium Ion
The hadronic properties of the pionium ion (Coulomb bound system of three
charged pions) are estimated using the results for the positronium ion
. It turns out that the hadronic shift of the ground state energy and
the lifetime of the pionium ion are approximately the same as for pionium.Comment: RevTex, 5 page
Majorana solutions to the two-electron problem
A review of the known different methods and results devised to study the
two-electron atom problem, appeared in the early years of quantum mechanics, is
given, with particular reference to the calculations of the ground state energy
of helium. This is supplemented by several, unpublished results obtained around
the same years by Ettore Majorana, which results did not convey in his
published papers on the argument, and thus remained unknown until now.
Particularly interesting, even for current research in atomic and nuclear
physics, is a general variant of the variational method, developed by Majorana
in order to take directly into account, already in the trial wavefunction, the
action of the full Hamiltonian operator of a given quantum system. Moreover,
notable calculations specialized to the study of the two-electron problem show
the introduction of the remarkable concept of an effective nuclear charge
different for the two electrons (thus generalizing previous known results), and
an application of the perturbative method, where the atomic number Z was
treated effectively as a continuous variable, contributions to the ground state
energy of an atom with given Z coming also from any other Z. Instead,
contributions relevant mainly for pedagogical reasons count simple broad range
estimates of the helium ionization potential, obtained by suitable choices for
the wavefunction, as well as a simple alternative to Hylleraas' method, which
led Majorana to first order calculations comparable in accuracy with well-known
order 11 results derived, in turn, by Hylleraas.Comment: amsart, 20 pages, no figure
Sharp regularity results for many-electron wave functions
We show that electronic wave functions Psi of atoms and molecules have a
representation Psi=F*phi, where F is an explicit universal factor, locally
Lipschitz, and independent of the eigenvalue and the solution Psi itself, and
phi has locally bounded second derivatives. This representation turns out to be
optimal as can already be demonstrated with the help of hydrogenic wave
functions. The proofs of these results are, in an essential way, based on a new
elliptic regularity result which is of independent interest. Some identities
that can be interpreted as cusp conditions for second order derivatives of Psi
are derived.Comment: 43 page
Nonrelativistic ionization energy for the helium ground state
The helium ground state nonrelativistic energy with 24 significant digits is
presented. The calculations are based on variational expansion with randomly
chosen exponents. This data can be used as a benchmark for other approaches for
many electron and/or three-body systems.Comment: 3 pages, 0 figure
Determination of a Wave Function Functional
In this paper we propose the idea of expanding the space of variations in
standard variational calculations for the energy by considering the wave
function to be a functional of a set of functions , rather than a function. In this manner a greater flexibility to
the structure of the wave function is achieved. A constrained search in a
subspace over all functions such that the wave function functional
satisfies a constraint such as normalization or the Fermi-Coulomb
hole charge sum rule, or the requirement that it lead to a physical observable
such as the density, diamagnetic susceptibility, etc. is then performed. A
rigorous upper bound to the energy is subsequently obtained by variational
minimization with respect to the parameters in the approximate wave function
functional. Hence, the terminology, the constrained-search variational method.
The \emph{rigorous} construction of such a constrained-search--variational wave
function functional is demonstrated by example of the ground state of the
Helium atom.Comment: 10 pages, 2 figures, changes made, references adde
About the stability of the dodecatoplet
A new investigation is done of the possibility of binding the "dodecatoplet",
a system of six top quarks and six top antiquarks, using the Yukawa potential
mediated by Higgs exchange. A simple variational method gives a upper bound
close to that recently estimated in a mean-field calculation. It is
supplemented by a lower bound provided by identities among the Hamiltonians
describing the system and its subsystems.Comment: 5 pages, two figures merged, refs. added, typos correcte
Weakly-Bound Three-Body Systems with No Bound Subsystems
We investigate the domain of coupling constants which achieve binding for a
3-body system, while none of the 2-body subsystems is bound. We derive some
general properties of the shape of the domain, and rigorous upper bounds on its
size, using a Hall--Post decomposition of the Hamiltonian. Numerical
illustrations are provided in the case of a Yukawa potential, using a simple
variational method.Comment: gzipped ps with 11 figures included. To appear in Phys. Rev.
Recurrence relations for four-electron integrals over Gaussian basis functions
In the spirit of the Head-Gordon-Pople algorithm, we report vertical,
transfer and horizontal recurrence relations for the efficient and accurate
computation of four-electron integrals over Gaussian basis functions. Our
recursive approach is a generalization of our algorithm for three-electron
integrals [J.~Chem.~Theory Comput.~12, 1735 (2016)]. The RRs derived in the
present study can be applied to a general class of multiplicative four-electron
operators. In particular, we consider various types of four-electron integrals
that may arise in explicitly-correlated F12 methods.Comment: 11 pages, 3 figures and 2 table
Dynamical stabilization of classical multi electron targets against autoionization
We demonstrate that a recently published quasiclassical M\oller type approach
[Geyer and Rost 2002, J. Phys. B 35 1479] can be used to overcome the problem
of autoionization, which arises in classical trajectory calculations for many
electron targets. In this method the target is stabilized dynamically by a
backward--forward propagation scheme. We illustrate this refocusing and present
total cross sections for single and double ionization of helium by electron
impact.Comment: LaTeX, 6 pages, 2 figures; submitted to J. Phys.
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