778 research outputs found
The self-energy of the uniform electron gas in the second order of exchange
The on-shell self-energy of the homogeneous electron gas in second order of
exchange, , is given by a certain integral. This integral is treated here in a
similar way as Onsager, Mittag, and Stephen [Ann. Physik (Leipzig) {\bf 18}, 71
(1966)] have obtained their famous analytical expression (in atomic units) for the correlation energy in
second order of exchange. Here it is shown that the result for the
corresponding on-shell self-energy is . The
off-shell self-energy correctly yields (the potential component of ) through the Galitskii-Migdal
formula. The quantities and appear in the
high-density limit of the Hugenholtz-van Hove (Luttinger-Ward) theorem.Comment: 12 pages, 2 figure
The on-shell self-energy of the uniform electron gas in its weak-correlation limit
The ring-diagram partial summation (or RPA) for the ground-state energy of
the uniform electron gas (with the density parameter ) in its
weak-correlation limit is revisited. It is studied, which treatment
of the self-energy is in agreement with the Hugenholtz-van
Hove (Luttinger-Ward) theorem and which is
not. The correlation part of the lhs h as the RPA asymptotics [in atomic units]. The use of renormalized RPA diagrams for the rhs
yields the similar expression with the sum rule
resulting from three sum rules for the components of and . This
includes in the second order of exchange the sum rule [P. Ziesche, Ann. Phys. (Leipzig), 2006].Comment: 19 pages, 10 figure
Reduced density matrices, their spectral resolutions, and the Kimball-Overhauser approach
Recently, it has been shown, that the pair density of the homogeneous
electron gas can be parametrized in terms of 2-body wave functions (geminals),
which are scattering solutions of an effective 2-body Schr\"odinger equation.
For the corresponding scattering phase shifts, new sum rules are reported in
this paper. These sum rules describe not only the normalization of the pair
density (similar to the Friedel sum rule of solid state theory), but also the
contraction of the 2-body reduced density matrix. This allows one to calculate
also the momentum distribution, provided that the geminals are known from an
appropriate screening of the Coulomb repulsion. An analysis is presented
leading from the definitions and (contraction and spectral) properties of
reduced density matrices to the Kimball-Overhauser approach and its
generalizations. Thereby cumulants are used. Their size-extensivity is related
to the thermodynamic limit.Comment: 15 pages, conference contributio
The 2-matrix of the spin-polarized electron gas: contraction sum rules and spectral resolutions
The spin-polarized homogeneous electron gas with densities
and for electrons with spin `up' () and spin `down'
(), respectively, is systematically analyzed with respect to its
lowest-order reduced densities and density matrices and their mutual relations.
The three 2-body reduced density matrices ,
, are 4-point functions for electron
pairs with spins , , and antiparallel,
respectively. From them, three functions ,
, , depending on only two variables,
are derived. These functions contain not only the pair densities but also the
1-body reduced density matrices. The contraction properties of the 2-body
reduced density matrices lead to three sum rules to be obeyed by the three key
functions , . These contraction sum rules contain corresponding
normalization sum rules as special cases. The momentum distributions
and , following from and
by Fourier transform, are correctly normalized through
. In addition to the non-negativity conditions
[these quantities are probabilities], it holds
and due to the Pauli principle and
due to the Coulomb repulsion. Recent parametrizations of the pair densities of
the spin-unpolarized homogeneous electron gas in terms of 2-body wave functions
(geminals) and corresponding occupancies are generalized (i) to the
spin-polarized case and (ii) to the 2-body reduced density matrix giving thus
its spectral resolutions.Comment: 32 pages, 4 figure
Methods for electronic-structure calculations - an overview from a reduced-density-matrix point of view
The methods of quantum chemistry and solid state theory to solve the
many-body problem are reviewed. We start with the definitions of reduced
density matrices, their properties (contraction sum rules, spectral
resolutions, cumulant expansion, -representability), and their determining
equations (contracted Schr\"odinger equations) and we summarize recent
extensions and generalizations of the traditional quantum chemical methods, of
the density functional theory, and of the quasi-particle theory: from finite to
extended systems (incremental method), from density to density matrix (density
matrix functional theory), from weak to strong correlation (dynamical mean
field theory), from homogeneous (Kimball-Overhauser approach) to inhomogeneous
and finite systems. Measures of the correlation strength are discussed. The
cumulant two-body reduced density matrix proves to be a key quantity. Its
spectral resolution contains geminals, being possibly the solutions of an
approximate effective two-body equation, and the idea is sketched of how its
contraction sum rule can be used for a variational treatment.Comment: 27 pages, conference contributio
The high-density electron gas: How its momentum distribution n(k) and its static structure factor S(q) are mutually related through the off-shell self-energy Sigma(k,omega)
It is shown {\it in detail how} the ground-state self-energy
of the spin-unpolarized uniform electron gas (with the
density parameter ) in its high-density limit determines: the
momentum distribution through the Migdal formula, the kinetic energy
from , the potential energy through the Galitskii-Migdal formula, the
static structure factor from by means of a Hellmann-Feynman
functional derivative. The ring-diagram partial summation or random-phase
approximation is extensively used and the results of Macke,
Gell-Mann/Brueckner, Daniel/Vosko, Kulik, and Kimball are summarized in a
coherent manner. There several identities were brought to the light.Comment: 34 pages, 6 figure
Designometry – Formalization of Artifacts and Methods
Two interconnected surveys are presented, one of artifacts and one of designometry. Artifacts are objects, which have an originator and do not exist in nature. Designometry is a new field of study, which aims to identify the originators of artifacts. The space of artifacts is described and
also domains, which pursue designometry, yet currently doing so without collaboration or common methodologies. On this basis, synergies as well as a generic axiom and heuristics for the quest of the creators of artifacts are introduced. While designometry has various areas of applications, the research of methods to detect originators of artificial minds, which constitute a subgroup of artifacts, can be seen as particularly relevant and, in the case of malevolent artificial minds, as contribution to AI safety
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