Shannon, Rényi, Tsallis Entropies and Onicescu Information Energy for Low-Lying Singly Excited States of Helium

Abstract

Knowledge of the electronic structures of atomic and molecular systems deepens our understanding of the desired system. In particular, several information-theoretic quantities, such as Shannon entropy, have been applied to quantify the extent of electron delocalization for the ground state of various systems. To explore excited states, we calculated Shannon entropy and two of its one-parameter generalizations, Rényi entropy of order α and Tsallis entropy of order α , and Onicescu Information Energy of order α for four low-lying singly excited states (1s2s 1 S e , 1s2s 3 S e , 1s3s 1 S e , and 1s3s 3 S e states) of helium. This paper compares the behavior of these three quantities of order 0.5 to 9 for the ground and four excited states. We found that, generally, a higher excited state had a larger Rényi entropy, larger Tsallis entropy, and smaller Onicescu information energy. However, this trend was not definite and the singlet–triplet reversal occurred for Rényi entropy, Tsallis entropy and Onicescu information energy at a certain range of order α .

Calculation of information entropies for the 1s2 state of helium‐like ions

“…To discuss the accuracy and the reliability of our findings using the proposed wavefunction of two parameters, we presented the twoelectron ground-state energies in Table 1. A comparison is made with literature in Table 1 for two-electron ground-state energies, as a function of Z, [16,31,33,34,52,53] and for He atom as a function of λ. [16,17,54,55] Compared with the available literature for He isoelectronic systems, for three parameters, our energies are in good agreement up to two digits for H − and increases to three digits as Z increases, for example, Be 2+ .…”

“…Note that, close to the critical value of λ, that is, λ $ λ c , the calculation of the energies and the information entropies will be questionable due to the inaccuracy of our numbers. [42] 0.52644 [52] 0.52775 [16,53] 0.1 2.5917 2.6031 [42] 2.60409 [16] 2.60444 [54] 2.6019 2.6135 [42] 2.61451 [16] 2.61485 [54] He 2.8911 2.9024 [42] 2.903724 [31,33,52] 2.903605 [34] 0.2 2.2959 2.3077 [42] 2.30876 [16] 2.30911 [17] 2.3330 2.3455 [42] 2.34666 [16] 2.34701 [54] 2.34750 [17] Li + 7.2682 7.2780 [42] 7.27948 7.27991 [16,33,53] 0.5 1.4571 1.4741 [42] 1.47695 [55] 1.6351 1.6525 [42] 1.60231 [8] 1.65540 [55] Be 2+ 13.6441 13.6531 [42] 13.6556 [53] 0.6 1.200652 1.2213 [42] 1.22499 [16] 1.4351 1.4549 [42] 1.45816 c…”

“…The potentials CP, YP, and ECSEP are used in calculating the information entropies in one‐ and two‐electron systems. [ 28–43 ] For two‐electron systems, the literature has used the density functional formalism [ 28 ] for Shannon entropy; many terms of the Hylleraas wavefunction [ 29–33 ] for Shannon, Rényi, and Tsallis entropies and Onicescu information energy; and configuration interaction [ 29,34 ] for linear and von Neumann entanglement entropies. Recently, two and three parameters have been used to calculate the information quantities in two‐electron systems, such as Fisher and Shannon, [ 44–46 ] Rényi, [ 39 ] and Tsallis.…”

“…The potentials CP, YP, and ECSEP are used in calculating the information entropies in one-and two-electron systems. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] For two-electron systems, the literature has used the density functional formalism [28] for Shannon entropy; many terms of the Hylleraas wavefunction [29][30][31][32][33] for Shannon, Rényi, and Tsallis entropies and Onicescu information energy; and configuration interaction [29,34] for linear and von Neumann entanglement entropies.…”

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

“…[32][33][34][35]. Unique properties of the Rényi and Tsallis entropies explain their wide applications in almost every branch of science and other fields of human acitivity: from seismology [36] and ecology [37,38] with geography [39] through medicine [40,41] and biology [42] to quantum physics [43][44][45][46][47][48][49][50][51][52], free field theories [53,54], and astronomy [55], with many, many others in between and beyond. Partially relevant to our discussion, let us point out that in the latest development, very recent experiments on Bose-Einstein condensate of interacting 87 Rb atoms loaded into a 1D [56] or 2D [57] optical lattice and on up to twenty 40 Ca + ions trapped into 1D straight string [58] directly measured the Rényi entanglement entropy with α = 2 of these many-body systems.…”