Information entropy and complexity measure in generalized Kratzer potential

Abstract

Shannon entropy (S), Fisher information (I) and a measure equivalent to Fisher-Shannon complexity (C IS ) of a ro-vibrational state of diatomic molecules (O 2 , O + 2 , NO, NO + ) with generalized Kratzer potential is analyzed. Exact analytical expression of I r is derived for the arbitrary state, whereas the same could be done for I p with {n, ℓ, m = 0} state. It is found that shifting from neutral to the cationic system, I r increases while S r decreases, consistent with the interpretation of a localization in the probability distribution. Additionally, this study reveals that C IS increases with the number of nodes in a system.Ever since the early days of quantum mechanics, diatomic molecular potentials have received much attention due to their importance to describe intra-molecular and intermolecular interactions as well as atomic-pair correlations. Over the years, a large number of such potentials have been adopted in a multitude of physical/chemical problems. Since the literature is quite vast, here we mention a few prominent ones, such as generalized Morse, Mie, Kratzer-Fues, pseudoharmonic, non-central, deformed Rosen-Morse, generalized Woods-Saxon, Pöschl-Teller potential etc [1]. They have fundamental relevance and utility in quantum description of natural phenomena, processes and systems, not only in the 3D world, but also in non-relativistic and relativistic D-dimensional physics. This work focuses on generalized Kratzer (Kratzer-type) potential, which is a simple realistic zero-order model for describing the vibration-rotation motion of diatomic molecules. It has importantproperties like (i) correct asymptotic behaviour at r = 0 and r = ∞ (ii) exactly solvable for a given state with an arbitrary angular momentum (iii) allows the system to dissociate, which is forbidden for a harmonic oscillator-based model. The mixed energy spectrum of this potential contains both discrete and continuum parts, corresponding to bound and scattering states respectively. While bound-state wave functions have been widely used in studies related to molecular spectroscopy, latter states are important in the context of the photo-dissociation process, diatomic molecular scattering, radiative recombination in an atom-atom collision [2], etc. Recently its use as a universal potential for diatomic molecules has been reported in two excellent review articles [3,4]. Apart from these, this has also been heavily studied in molecular, chemical and solid-state physics, due to its general feature of true interaction energy as well as interatomic, inter-molecular and dynamical properties [5][6][7].In recent years, there has been a burgeoning activity in studies related to informationtheoretic measures in quantum chemical systems. The delocalizing properties of electronic distribution that characterize the quantum states of these molecular potentials have been analyzed quite extensively by information theoretical tools such as Shannon entropy (S) [8], Fisher information (I) [9], Rényi entropy (R) [10], Tsallis entropy (T ) [11] and Onic...

Shannon entropy and Fisher information for screened Kratzer potential

“…The wavefunction is expressed as [ 12 ] where Ω represents a solid angle derived from the quantum numbers n , l , m . The normalization condition is obtained as [ 53 ] …”

“…[ 8–10 ] The significance of the global measure is to investigate the uncertainty associated with the probability distribution. [ 11–14 ] The position and momentum spaces of the Shannon entropy have an entropic relation derived by Berkner, Bialynicki‐Birula, and Mycieslki [ 15 ] and expressed as S r + S p ≥ D (1 + ln π ), where D is the spatial dimension. This relation is proven to be superior to the Heinsberg uncertainty relation as it can account for a higher order.…”

“…[ 18–20 ] The study of Fisher information finds its significance in density functional. [ 21–23 ] Fisher information entropy, defined as probability density in position and momentum spaces, is expressed as follows [ 11,12 ] : …”

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

“…Originated from the information theory and later extended to the position space, the information-theoretic (IT) quantities provide a possible quantitative account of the extent of electron localization. For instance, the more delocalized is the electron distribution, the larger is the Shannon entropy (S) [3][4][5][6][7][8][9][10][11][12][13][14]. Moreover, Rényi entropy of order α (R α ) [15][16][17] and Tsallis entropy of order α (T α ) [18][19][20][21] are the one-parameter generalizations of Shannon entropy and offer extended descriptions.…”

“…In addition, Onicescu Information Energy of order α (E O α ) [22,23] is related to the frequency moment of the electron density [24][25][26]. These IT quantities have been applied to various atomic and molecular systems [4][5][6][7][8][9][10][11][12][13][14] and potentials [81][82][83][84][85][86][87][88][89][90][91]. Further background details of these IT quantities were summarized by Ilić and Stanković [92] and by us [80].…”