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Analysis of Information Entropies for He-Like Ions

Hamid Al-Jibbouri

Abstract


The electronic structure, a special quality of an atomic or molecular system, is the major factor for further realization of physical results. However, in this paper, we present the topical issue of normalized electron density in position and momentum spaces, Shannon, Rényi, and Tsallis entropies to quantify the reach of electron delocalization for several atomic systems. Hartree-Fock-Roothaan (HFR) wave function is performed and considered for He-like ions using single-Zeta 𝛽-type orbital (βTOs) basis set to investigate the affecting of electron density and information entropies. The electron density maxima in position space are raised, and their positions move toward the nucleus as Z increases, in accordance with the increasing attractive force of the nucleus, and vice versa in momentum space. Shannon’s entropy has impacted the delocalization of the electron in different atomic systems. In the limit γ→1, both Rényi and Tsallis entropy results recover Shannon’s entropy value. Rényi and Tsallis entropies decrease by increasing γ. Indeed, the estimated results have been calculated via the Wolfram Mathematica program and have good agreement with the literature results. The obtained results may be a useful reference for future studies on theoretical information quantities.

 

Doi: 10.28991/ESJ-2022-06-04-08

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Keywords


Shannon; Rényi; Tsallis Entropies; Hartree-Fock-Roothaan; Helium-Isoelectronic Series; 𝛽-type Orbitals.

References


Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423. doi:10.1002/j.1538-7305.1948.tb01338.x.

Rényi, A. (1961, January). On measures of entropy and information. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, 4, 547-562, University of California Press, Oakland, United States.

Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52(1–2), 479–487. doi:10.1007/BF01016429.

Hô, M., Sagar, R. P., Pérez-Jordá, J. M., Smith, V. H., & Esquivel, R. O. (1994). A numerical study of molecular information entropies. Chemical Physics Letters, 219(1–2), 15–20. doi:10.1016/0009-2614(94)00029-8.

Saha, S., & Jose, J. (2020). Shannon entropy as a predictor of avoided crossing in confined atoms. International Journal of Quantum Chemistry, 120(22), e26374. doi:10.1002/qua.26374.

Martínez-Flores, C. (2021). The information theory of the helium atom in screened Coulomb potentials. International Journal of Quantum Chemistry, 121(6), e26529. doi:10.1002/qua.26529.

Nasser, I., Zeama, M., & Abdel-Hady, A. (2019). The nonadditive entropy for the ground state of helium-like ions using Hellmann potential. Molecular Physics, 118(3), 1612105. doi:10.1080/00268976.2019.1612105.

Nasser, I., Zeama, M., & Abdel-Hady, A. (2020). Rényi, Fisher, Shannon, and their electron correlation tools for two-electron series. Physica Scripta, 95(9), 095401. doi:10.1088/1402-4896/abaa09.

Lin, Y. C., Lin, C. Y., & Ho, Y. K. (2013). Spatial entanglement in two-electron atomic systems. Physical Review A - Atomic, Molecular, and Optical Physics, 87(2), 022316. doi:10.1103/PhysRevA.87.022316.

Lin, Y. C., & Ho, Y. K. (2014). Quantum entanglement for two electrons in the excited states of helium-like systems. Canadian Journal of Physics, 93(6), 646–653. doi:10.1139/cjp-2014-0437.

Sen, K. D. (2005). Characteristic features of Shannon information entropy of confined atoms. Journal of Chemical Physics, 123(7), 074110. doi:10.1063/1.2008212.

Nahum, A., Ruhman, J., & Huse, D. A. (2018). Dynamics of entanglement and transport in one-dimensional systems with quenched randomness. Physical Review B, 98(3), 035118. doi:10.1103/PhysRevB.98.035118.

Barghathi, H., Herdman, C. M., & Del Maestro, A. (2018). Rényi Generalization of the Accessible Entanglement Entropy. Physical Review Letters, 121(15), 150501. doi:10.1103/PhysRevLett.121.150501.

Herdman, C. M., Roy, P. N., Melko, R. G., & Maestro, A. Del. (2017). Entanglement area law in superfluid 4 He. Nature Physics, 13(6), 556–558. doi:10.1038/nphys4075.

Kaufman, A. M., Tai, M. E., Lukin, A., Rispoli, M., Schittko, R., Preiss, P. M., & Greiner, M. (2016). Quantum thermalization through entanglement in an isolated many-body system. Science, 353(6301), 794–800. doi:10.1126/science.aaf6725.

Rouse, I., & Willitsch, S. (2017). Superstatistical Energy Distributions of an Ion in an Ultracold Buffer Gas. Physical Review Letters, 118(14), 143401. doi:10.1103/PhysRevLett.118.143401.

Dechant, A., Kessler, D. A., & Barkai, E. (2015). Deviations from Boltzmann-Gibbs statistics in confined optical lattices. Physical Review Letters, 115(17), 173006. doi:10.1103/PhysRevLett.115.173006.

Najafizade, S. A., Hassanabadi, H., & Zarrinkamar, S. (2016). Nonrelativistic Shannon information entropy for Kratzer potential. Chinese Physics B, 25(4), 40301. doi:10.1088/1674-1056/25/4/040301.

Amadi, P. O., Ikot, A. N., Ngiangia, A. T., Okorie, U. S., Rampho, G. J., & Abdullah, H. Y. (2020). Shannon entropy and Fisher information for screened Kratzer potential. International Journal of Quantum Chemistry, 120(14), e26246. doi:10.1002/qua.26246.

Dehesa, J. S., Koga, T., Yáñez, R. J., Plastino, A. R., & Esquivel, R. O. (2011). Quantum entanglement in helium. Journal of Physics B: Atomic, Molecular and Optical Physics, 45(1), 015504. doi:10.1088/0953-4075/45/1/015504.

López-Rosa, S., Martín, A. L., Antolín, J., & Angulo, J. C. (2019). Electron-pair entropic and complexity measures in atomic systems. International Journal of Quantum Chemistry, 119(7), e25861. doi:10.1002/qua.25861.

Lin, C. H., & Ho, Y. K. (2015). Shannon information entropy in position space for two-electron atomic systems. Chemical Physics Letters, 633(5), 261–264. doi:10.1016/j.cplett.2015.05.029.

Nascimento, W. S., de Almeida, M. M., & Prudente, F. V. (2021). Coulomb correlation and information entropies in confined helium-like atoms. European Physical Journal D, 75(6). doi:10.1140/epjd/s10053-021-00177-6.

Al-Jibbouri, H. Compton Profile of 1s2-State for 2≤Z≤10. Indian Journal of Pure & Applied Physics, 59(11), 752–755. Available online: Available online: http://nopr.niscair.res.in/handle/123456789/58624 (accessed on January 2022).

Majumdar, S., & Roy, A. K. (2020). Shannon entropy in confined he-like ions within a density functional formalism. Quantum Reports, 2(1), 189–207. doi:10.3390/quantum2010012.

Ou, J. H., & Ho, Y. K. (2019). Benchmark calculations of Rényi, Tsallis entropies, and Onicescu information energy for ground state helium using correlated Hylleraas wave functions. International Journal of Quantum Chemistry, 119(14), e25928. doi:10.1002/qua.25928.

Ou, J. H., & Ho, Y. K. (2017). Shannon information entropy in position space for the ground and singly excited states of helium with finite confinements. Atoms, 5(2), 15. doi:10.3390/atoms5020015.

Toranzo, I. V., Puertas-Centeno, D., & Dehesa, J. S. (2016). Entropic properties of D-dimensional Rydberg systems. Physica A: Statistical Mechanics and Its Applications, 462(6), 1197–1206. doi:10.1016/j.physa.2016.06.144.

Estañón, C. R., Aquino, N., Puertas-Centeno, D., & Dehesa, J. S. (2020). Two-dimensional confined hydrogen: An entropy and complexity approach. International Journal of Quantum Chemistry, 120(11), e26192. doi:10.1002/qua.26192.

Liu, S.-B., Rong, C.-Y., Wu, Z.-M., & Lu, T. (2015). Rényi Entropy, Tsallis Entropy and Onicescu Information Energy in Density Functional Reactivity Theory. Acta Physico-Chimica Sinica, 31(11), 2057–2063. doi:10.3866/pku.whxb201509183.

Ou, J. H., & Ho, Y. K. (2019). Shannon, rényi, tsallis entropies and onicescu information energy for low-lying singly excited states of helium. Atoms, 7(3), 1–15. doi:10.3390/atoms7030070.

Romera, E., & Dehesa, J. S. (2004). The Fisher-Shannon information plane, an electron correlation tool. Journal of Chemical Physics, 120(19), 8906–8912. doi:10.1063/1.1697374.

Shi, Q., & Kais, S. (2004). Finite size scaling for the atomic Shannon-information entropy. Journal of Chemical Physics, 121(12), 5611–5617. doi:10.1063/1.1785773.

Shi, Q., & Kais, S. (2005). Discontinuity of Shannon information entropy for two-electron atoms. Chemical Physics, 309(2–3), 127–131. doi:10.1016/j.chemphys.2004.08.020.

Farid, M., Abdel-Hady, A., & Nasser, I. (2017). Comparative study of the scaling behavior of the Rényi entropy for He-like atoms. Journal of Physics: Conference Series, 869, 012011. doi:10.1088/1742-6596/869/1/012011.

Nasser, I., Zeama, M., & Abdel-Hady, A. (2017). The Rényi entropy, a comparative study for He-like atoms using the exponential-cosine screened Coulomb potential. Results in Physics, 7(2), 3892–3900. doi:10.1016/j.rinp.2017.10.013.

Nasser, I., Zeama, M., & Abdel-Hady, A. (2021). Calculation of information entropies for the 1s2 state of helium-like ions. International Journal of Quantum Chemistry, 121(5), e26499. doi:10.1002/qua.26499.

Zeama, M., & Nasser, I. (2019). Tsallis entropy calculation for non-Coulombic helium. Physica A: Statistical Mechanics and Its Applications, 528, 121468. doi:10.1016/j.physa.2019.121468.

Onate, C. A., Ikot, A. N., Onyeaju, M. C., Ebomwonyi, O., & Idiodi, J. O. A. (2018). Effect of dissociation energy on Shannon and Rényi entropies. Karbala International Journal of Modern Science, 4(1), 134–142. doi:10.1016/j.kijoms.2017.12.004.

Ema, I., De La Vega, J. M. G., Miguel, B., Dotterweich, J., Meißner, H., & Steinborn, E. O. (1999). Exponential-type basis functions: Single- and double-zeta B function basis sets for the ground states of neutral atoms from Z = 2 to Z = 36. Atomic Data and Nuclear Data Tables, 72(1), 57–99. doi:10.1006/adnd.1999.0809.

Atkins, P. W., & Friedman, R. S. (2011). Molecular quantum mechanics (4th Ed.). Oxford University Press, Oxford, United States.

Filter, E., & Steinborn, E. O. (1978). Extremely compact formulas for molecular two-center one-electron integrals and Coulomb integrals over Slater-type atomic orbitals. Physical Review A, 18(1), 1–11. doi:10.1103/PhysRevA.18.1.

Ertürk, M., & Sahin, E. (2020). Generalized B functions applied to atomic calculations. Chemical Physics, 529, 110549. doi:10.1016/j.chemphys.2019.110549.

Al-Jibbouri, H. (2019). Ground State of Radial-Radial Distribution Function for C+4 and O+6 Ions. Journal of Physics: Conference Series, 1294(5), 052052. doi:10.1088/1742-6596/1294/5/052052.

Al-Jibbouri, H., & Alhasan, A. (2019). Study the Inter-Particle Function for Some Electronic System. Journal of Physics: Conference Series, 1294(2), 022014. doi:10.1088/1742-6596/1294/2/022014.

Mitnik, D. M., & Miraglia, J. E. (2005). Simple correlated wavefunctions for the K-shell electrons of neutral atoms. Journal of Physics B: Atomic, Molecular and Optical Physics, 38(18), 3325–3338. doi:10.1088/0953-4075/38/18/004.

Fernández Rico, J., López, R., Ramírez, G., & Ema, I. (1998). Multiple one-center expansions of charge distributions associated with Slater orbitals. Journal of Molecular Structure: THEOCHEM, 433(1–3), 7–18. doi:10.1016/S0166-1280(98)00005-0.

Gadre, S. R., Sears, S. B., Chakravorty, S. J., & Bendale, R. D. (1985). Some novel characteristics of atomic information entropies. Physical Review A, 32(5), 2602–2606. doi:10.1103/PhysRevA.32.2602.

Maassen, H., & Uffink, J. B. M. (1988). Generalized entropic uncertainty relations. Physical Review Letters, 60(12), 1103–1106. doi:10.1103/physrevlett.60.1103.

Panos, C. P., Nikolaidis, N. S., Chatzisavvas, K. C., & Tsouros, C. C. (2009). A simple method for the evaluation of the information content and complexity in atoms. A proposal for scalability. Physics Letters, Section A: General, Atomic and Solid State Physics, 373(27–28), 2343–2350. doi:10.1016/j.physleta.2009.04.070.

Al-Jibbouri, H. (2021). Variational calculation of lithium-like ions from B+2 to N+4 using β-type roothaan–hartree–fock wavefunction. Ukrainian Journal of Physics, 66(8), 684–690. doi:10.15407/ujpe66.8.684.

Saha, A., Talukdar, B., & Chatterjee, S. (2017). On the correlation measure of two-electron systems. Physica A: Statistical Mechanics and Its Applications, 474, 370–379. doi:10.1016/j.physa.2017.02.003.

Sagar, R. P., & Guevara, N. L. (2005). Mutual information and correlation measures in atomic systems. The Journal of Chemical Physics, 123(4), 044108. doi:10.1063/1.1953327.

Rastegin, A. E. (2014). Uncertainty and certainty relations for Pauli observables in terms of Rényi entropies of order α ∈ (0; 1]. Communications in Theoretical Physics, 61(3), 293–298. doi:10.1088/0253-6102/61/3/04.

Delgado-Soler, L., Toral, R., Tomás, M. S., & Rubio-Martinez, J. (2009). RED: A set of molecular descriptors based on rényi entropy. Journal of Chemical Information and Modeling, 49(11), 2457–2468. doi:10.1021/ci900275w.

Toranzo, I. V., & Dehesa, J. S. (2016). Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems. EPL (Europhysics Letters), 113(4), 48003. doi:10.1209/0295-5075/113/48003.


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DOI: 10.28991/ESJ-2022-06-04-08

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