Correlation strength and information entropy

Ziesche, P.

doi:10.1002/qua.560560422

Cited by 94 publications

(91 citation statements)

References 34 publications

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“…Relations with the correlation strength have been also established by Ziesche [5], while many other works have been focused on the link between the nonidempotency of the one-particle density matrix and complexity measures (see, for example, Nagy and Romera [6] and other references therein). In a slightly different context, Romera and Dehesa [7] have introduced a rather sophisticated measure of the electron correlation based on the combination of Shannon and Fischer information.…”

Section: Introductionmentioning

confidence: 89%

Shannon Entropy in Atoms: A Test for the Assessment of Density Functionals in Kohn-Sham Theory

Amovilli

Floris

2018

Computation

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Electron density is used to compute Shannon entropy. The deviation from the Hartree-Fock (HF) of this quantity has been observed to be related to correlation energy. Thus, Shannon entropy is here proposed as a valid quantity to assess the quality of an energy density functional developed within Kohn-Sham theory. To this purpose, results from eight different functionals, representative of Jacob's ladder, are compared with accurate results obtained from diffusion quantum Monte Carlo (DMC) computations. For three series of atomic ions, our results show that the revTPSS and the PBE0 functionals are the best, whereas those based on local density approximation give the largest discrepancy from DMC Shannon entropy.

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Section: Introductionmentioning

confidence: 89%

Shannon Entropy in Atoms: A Test for the Assessment of Density Functionals in Kohn-Sham Theory

Amovilli

Floris

2018

Computation

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“…The point to be remembered, however, is that this information loss is being driven by the inter-particle interactions in the system while the corresponding information loss in a thermodynamic system is driven by its finite temperature, and the thermal fluctuations caused by it. In various contexts, other authors have also computed and discussed the information entropy associated with interacting quantum systems [21][22][23].…”

Section: Theorymentioning

confidence: 99%

A basis-set based Fortran program to solve the Gross–Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps

Tiwari¹,

Shukla²

2006

Computer Physics Communications

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Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schrödinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simpleharmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas-Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way. Program summary Method of solution:The solutions of the Gross-Pitaevskii equation corresponding to the condensates in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Gross-Pitaevskii equation which is a second-order nonlinear differential equation, is transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra. Unusual features of the program: None

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“…[8,9,10,11,12,13,14,15] In Refs. [16,17], these concepts were extended to mixed states, which requires the full density matrix instead of the 1PDM.…”

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confidence: 99%

Physics behind the minimum of relative entropy measures for correlations

Held

Mauser²

2013

Eur. Phys. J. B

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Abstract. The relative entropy of a correlated state and an uncorrelated reference state is a reasonable measure for the degree of correlations. A key question is however which uncorrelated state to compare to. The relative entropy becomes minimal for the uncorrelated reference state that has the same oneparticle density matrix as the correlated state. Hence, this particular measure, coined nonfreeness, is unique and reasonable. We demonstrate that for relevant physical situations, such as finite temperatures or a correlation enhanced orbital splitting, other choices of the uncorrelated state, even educated guesses, overestimate correlations. e.g. in high-temperature superconductors. However, correlations are particularly difficult to deal with in theory and even a universally agreed definition, or measure, of "correlation" is hitherto lacking. There is a general agreement that a Hartree-Fock Slater determinant represents an uncorrelated state, even though such a wave function includes formally something one might call "correlations", which originate from the antisymmetrization of the wave function. Hence a correlation measure typically considers the difference of the correlated state vs. an uncorrelated Hartree-Fock calculation. In this situation, the questions are: For what quantity should one consider the difference? To which uncorrelated (possibly mixed) state should one compare to?One possibility to quantify correlation is to look at the energy differencebetween the (correlated) state investigated (E corr ) and an uncorrelated (or free) state E free , which is also coined correlation energy. This is e.g. the typical quantity considered in quantum chemistry or density functional theory (DFT) [4,5,6]. The exchange-correlation energy E xc is, as the difference to the Hartree energy, also readily accessible in DFT, at least within e.g. the local density approximation (LDA). This requires, however, still a separation into exchange and correlation part. In many-body theory on the other hand, one often considers two particle correlation functions of the type [7]

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Correlation strength and information entropy

Cited by 94 publications

References 34 publications

Shannon Entropy in Atoms: A Test for the Assessment of Density Functionals in Kohn-Sham Theory

Shannon Entropy in Atoms: A Test for the Assessment of Density Functionals in Kohn-Sham Theory

A basis-set based Fortran program to solve the Gross–Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps

Physics behind the minimum of relative entropy measures for correlations

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