The He isoelectronic series and the Hooke’s law model: Correlation measures and modifications of Collins’ conjecture

Ziesche, P.; Smith, Vedene H.; Hô, Minhhuy; Rudin, Sven P.; Gersdorf, Peter; Taut, M.

doi:10.1063/1.478560

Cited by 75 publications

(60 citation statements)

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“…The actual strength of correlations in a given system is usually quantified by an interaction strength U relative to an energy unit such as the bandwidth W , or by comparing expectation values of particular operators, e.g., the interaction or the total energy [6][7][8][9][10]. However, there are many other quantities which can in principle be used to measure the correlation strength.…”

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confidence: 99%

Quantification of Correlations in Quantum Many-Particle Systems

et al. 2012

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We introduce a well-defined and unbiased measure of the strength of correlations in quantum many-particle systems which is based on the relative von Neumann entropy computed from the density operator of correlated and uncorrelated states. The usefulness of this general concept is demonstrated by quantifying correlations of interacting electrons in the Hubbard model and in a series of transition-metal oxides using dynamical mean-field theory.PACS numbers: 74.70. Tx, 71.10.Fd, 71.30.+h Correlations in solids are the origin of many surprising phenomena such as Mott-insulating behavior and hightemperature superconductivity [1][2][3]. During the last few years the investigation of correlations between ultracold atoms in optical lattices has opened another fascinating field of research [4].In many-body physics correlations are conventionally defined as the effects which go beyond factorization approximations such as Hartree-Fock theory [5]. The actual strength of correlations in a given system is usually quantified by an interaction strength U relative to an energy unit such as the bandwidth W , or by comparing expectation values of particular operators, e.g., the interaction or the total energy [6][7][8][9][10]. However, there are many other quantities which can in principle be used to measure the correlation strength. Indeed "correlations" are by definition a relative concept since they always require the comparison with some reference system. Any approach which employs the expectation value of a particular set of operators for comparison will be biased. This raises a fundamental question: Is there an objective way to quantify the correlations of a system, which even allows one to compare the correlation strength of different systems?Maximal information about a general quantum state is provided by the corresponding density operator ("statistical operator")ρ = i p i |ψ i ψ i |, where p i is the probability for the quantum state |ψ i to be present in the mixture [11]. If there exists a basis in whichρ can be completely factorized (ρ →ρ PS , where PS refers to "product state"), the corresponding state is by definition uncorrelated [12]. To quantify the correlation strength of a quantum state one has to compareρ withρ PS in a suitable way.In this Letter we propose to quantify correlations within a statistical approach which is based on the concept of the von Neumann entropy [11,13]. We will show that the relative entropy [14] of a quantum state with respect to an uncorrelated product state provides a well-defined, unbiased and useful measure of the correlation strength in that system. This concept not only allows one to quantify correlations, but even to compare the correlation strength of different quantum states. In quantum theory the von Neumann entropy is defined as S(ρ) = − lnρ ρ = −Tr(ρ lnρ). It quantifies the degree to which a quantum system is in a mixed state. In this way one can also define a relative entropy [14]provided the support [15] ofρ is contained in the support ofσ; otherwise ∆S(ρ||σ) is infinite. ...

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confidence: 99%

Quantification of Correlations in Quantum Many-Particle Systems

et al. 2012

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“…In applicative context it will, therefore, be quite interesting to examine how S and F respond to important physical effects, like the electron-electron correlation, which plays a role in the physics of many-electron systems such as atoms, molecules and solids. There exists a large number of studies [4] which attempted to establish that correlation plays a more dominant role in modifying the bare values of S than it does in correcting the Hartree-Fock binding energies of atomic systems. In particular, by working with a few ions in the Helium iso-electronic sequence, Romera and Dehesa [5] introduced the product of the position-space Shannon entropy power and Fisher information as correlation measure of two-electron systems.…”

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confidence: 99%

Shannon entropies and Fisher information of K-shell electrons of neutral atoms

Sekh¹,

Saha

Talukdar

2018

Physics Letters A

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We represent the two K-shell electrons of neutral atoms by Hylleraas-type wave function which fulfils the exact behavior at the electron-electron and electron-nucleus coalescence points and, derive a simple method to construct expressions for single-particle position-and momentum-space charge densities, ρ(r) and γ(p) respectively. We make use of the results for ρ(r) and γ(p) to critically examine the effect of correlation on bare (uncorrelated) values of Shannon information entropies (S) and of Fisher information (F ) for the K-shell electrons of atoms from helium to neon. Due to inter-electronic repulsion the values of the uncorrelated Shannon position-space entropies are augmented while those of the momentum-space entropies are reduced. The corresponding Fisher information are found to exhibit opposite behavior in respect of this. Attempts are made to provide some plausible explanation for the observed response of S and F to electronic correlation.

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“…correlation energy and entropy [36][37][38][39][40]. This idea underlies several recent studies, including that by Delle Site that also uses Shannon entropic indices albeit directly on the electron density [41][42][43].…”

Section: Theoretical Aspectsmentioning

confidence: 97%