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International Journal of Quantum Chemistry

A relationship between atomic correlation energy of neutral atoms and generalized entropy

Alberto Grassi

Published Mar 25, 2010• 27 citations
Publisher

Abstract

Abstract In a previous article, hereafter named as Paper I, we have showed a relationship between atomic correlation energy of neutral atoms with 2 < Z < 29 and Tsallis entropy. In this article, we generalize this relation showing the link between the atomic correlation energy and a general form of entropy obtained from deformed algebra. The results evidence the role of both q and Δ parameters of the general entropy, in terms of contribution of the long‐range interactions in the correlation energy. The q and Δ values, obtained as best fit of the atomic correlation energies 2 < Z < 29, indicate that this general form of entropy reduces to the Tsallis one, reproducing well the trend of the correlation energy for low Z. Moreover, as a consequence of these values of the parameters, the state atomic wave function is more localized with respect to the wave function calculated in the limit of Shannon entropy. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

DOI
10.1002/qua.22541

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Abstract In a previous article, hereafter named as Paper I, we have showed a relationship between atomic correlation energy of neutral atoms with 2 < Z < 29 and Tsallis entropy. In this article, we generalize this relation showing the link between the atomic correlation energy and a general form of entropy obtained from deformed algebra. The results evidence the role of both q and Δ parameters of the general entropy, in terms of contribution of the long‐range interactions in the correlation energy. The q and Δ values, obtained as best fit of the atomic correlation energies 2 < Z < 29, indicate that this general form of entropy reduces to the Tsallis one, reproducing well the trend of the correlation energy for low Z. Moreover, as a consequence of these values of the parameters, the state atomic wave function is more localized with respect to the wave function calculated in the limit of Shannon entropy. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
In this report, we present three generalized expressions of Shannon’s entropy using the electron densities of position and momentum spaces. Such expressions were obtained using the definition of deformed logarithm introduced by Tsallis. We show that only one expression fulfils the dimensionless criterion, it is strictly positive overall space and follows a growing behavior with respect to the electron number. We also found that by using some values of [Formula: see text], it is possible to reproduce the general trends of the radial distribution in position space and momentum space of the neon atom.
In this paper, an analytical model of 2p orbital helium atom has been considered to quantify the values of the Rényi and Tsallis entropies along with their theoretical aspects. The normalized radial wave function used here (in atomic units) is obtained by solving the Schrödinger equation. The complete form of the coordinate space wave function is obtained by the use of the spherical harmonics. The momentum space wave function is obtained by taking the Fourier transform of the coordinate space wave function. The probability densities constituted with the respective coordinate and momentum space wave functions have been used to compute the numerical values of the Rényi and Tsallis entropies in the coordinate and momentum space for different values of the order β varying from 2 to 10. The computed values are presented in a tabular form. Further, it is mathematically demonstrated that in the limit of order β → 1, both the Rényi and Tsallis entropies lead to the Shannon entropy. Finally, an outlook of the present work has been summarized with some concluding remarks.
We calculate the analytical and numerical values of the position space Shannon entropy, momentum space Shannon entropy, and total Shannon entropy, [Formula: see text], [Formula: see text], and [Formula: see text], respectively, of free and trapped Rydberg hydrogen-like atoms. The influence of atomic number Z, the principal quantum number n, and energy E on the Shannon entropy of the Rydberg atoms are illustrated. The scaling properties of Shannon entropy with energy of states E and the principal quantum number n have been reported for the first time to the best of our knowledge. Our work explains how Shannon entropy indicates localization-delocalization of the wavefunction. The total Shannon entropy as a measure of the number of nodes in the trapped Rydberg atom's wavefunction is also discussed. We show why an uncertainty relation based on Shannon entropy is superior to Heisenberg uncertainty for Rydberg atoms.
Abstract The correlation present in the nondegenerate ground state of an interacting Fermi system is discussed in terms of reduced density matrices and their cumulant expansion. By generalizing a result obtained for the interacting uniform electron gas (correlation induced exchange‐hole narrowing), possible measures of the correlation strength in terms of natural occupation numbers (the eigenvalues of the true one‐particle density matrix) are introduced. These quantities‐thev‐order nonidempotency and the information entropy of the natural occupation numbers‐result from the correlated many‐body wave function and characterize the ground‐state correlation in addition to the usual correlation energy. The uniform electron gas serves as a first illustrative example. © 1995 John Wiley & Sons, Inc.
The universal property for the information entropy S = a + h In Ζ is verified for atoms using a systematic study with Roothaan-Hartree-Fock (RHF) wave functions with atomic number Ζ — 2 — 54. The above relation was proposed previously for atoms, nuclei, atomic clusters and correlated atoms in a trap. Kullback-Leibler relative entropy Κ and Jensen-Shannon divergence J are employed to compare RHF with Thomas-Fermi (TF) density of atoms as well as another phenomenological density obtained by Sagar et al. Two-body density distributions in position- and momentum-space are used to calculate and compare the corresponding information entropies for correlated and uncorrelated nuclei and bosonic systems (correlated atoms in a trap). It is seen that short-range correlations (SRC) increase the values of S. One-body information entropy entropy S\ is compared with two-body information entropy and a conjecture is made for TV-body information entropy SN- The entropy Κ and the divergence J are also used to evaluate the information distance between correlated and uncorrelated densities both at the one- and the two-body levels for nuclei and trapped Bose gases.