Electron-pair entropic and complexity measures in atomic systems

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ Esta es la versión aceptada del artículo publicado en: This is a accepted manuscript of a paper published in: International Journal of Quantum Chemistry, vol. 119, issue 7, e25861 DOI: https://doi.org/10.1002/qua.25861 Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. “This is the peer reviewed version of the following article: Electron-pair entropic and complexity measures in atomic systems, which has been published in final form at https://doi.org/10.1002/qua.25861. This article may be used for noncommercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited." Electron-pair entropic and complexity measures in atomic systems S. L´opez-Rosa∗ , A.L. Mart´ın† , J. Antol´ın ‡ , J.C. Angulo§ October 25, 2018 Abstract The two-electron atomic densities are analysed in both position and momentum spaces in terms of different information-theoretic measures, such as disequilibrium, Shannon entropy, shape complexity and its corresponding information plane. This study is conveyed throughout the Periodic Table and the obtained results are discussed in terms of varied atomic properties such as (i) atomic charge, (ii) shell filling patterns and (iii) electronic correlation. A detailed discussion on how these properties modify in a particular manner the electron density structure when considering a one-electron or a two-electron density description is conducted. ∗Departamento de F´ısica Aplicada II, Universidad de Sevilla, 41012-Sevilla, Spain. †Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Granada, 18071-Granada, Spain ‡Departamento de F´ısica Aplicada, Universidad de Zaragoza, 50018-Zaragoza, Spain. Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, 18071-Granada, Spain §Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Granada, 18071-Granada, Spain. Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, 18071-Granada, Spain 1 1INTRODUCTION One-electron densities have a straightforward meaning in atomic physics, as they are directly related to the probability of finding an electron in a determinate region of the atom. One can study different regions of the atomic density and thus one can understand the way electrons populating that region behave. There is a huge amount of information coded inside monoelectronic densities. However, there is some information that this kind of density lacks, mainly information about the electron correlations. These densities don’t directly give us any information at all about how the position of an electron conditions the position of the others. It is in this context when electron pair densities arise1. Nowadays Information Theory of quantum many-body systems is attracting the attention of scientists in several fields in physical sciences, wherein major areas of research are interconnected, i.e., physics, mathematics, chemistry, and biology. It is so that there is an inherent interest for applying information-theoretic ideas and methodologies to chemical, mesoscopic and biological systems along with the processes they are involved with. In line with the aforementioned developments, multidisciplinary research projects have been undertaken so as to employ Information Theory at different levels, classical (Shannon, Fisher, complexity, etc) and quantum (von Neumann and other entanglement measures), on a variety of physical, chemical and biological systems and processes2–4. The Information Theory of quantum systems provides an entropy-based characterization of the atomic and molecular systems, which complements the energy-based representation obtained with the wave function and density functional methods. The physical and chemical properties of these systems can be described by means of spreading measures of entropic character of the electron density5,6. These measures of uncertainty, randomness, disorder and localization are basic ingredients encountered to play a relevant role for the identification and description of numerous quantum phenomena in physical systems and chemical processes. The magnitudes of the Information Theory are very useful when trying to understand different traits and behaviours of atomic systems. There have been plenty of studies of quantum systems by means of the informational measures7–9. Most of these studies have been focused on the monoelectronic distributions, providing analyses on, e.g., the Shannon 2 entropy10, Fisher information11, similarity indices12,13, different divergence measures14,15 and complexity measures16–18. However, as previously mentioned, studies focused on two-electron densities require another point of view in order to analyse correlation-like qualities19–24. In past years there have been some successful attempts to study the electron pair densities, exposing uncertainty relationships20, calculating information theoretical measures such as Shannon-related ones10,21,23 or similarity measures20,21,25,26. This knowledge gained over the electron pair densities has translated in numerous and diverse applications. Two-electron densities have been employed to the analysis and detection of chemical bonds in molecules, finding that regions where electrons presented a higher correlations were directly related to the position of the bonds27,28. Electron pair densities have also been employed as a scale-down method used to study many particle systems29–33. Even an alternative density functional theory has been developed, with electron pair density as the functional key1,34,35. Some of the analyses made in the past on the electron pair density could not be as exhaustive as would have been desirable due to technical limitations of the numerical methods used, unable to calculate the two electron densities for all the atomic systems in the Periodic Table. Preceeding studies on this subject are relatively recent36–39, as a consequence of dealing with a subject considered some years ago, but not efficiently developed until very recently. In fact, all those publications only deal with two-electron systems, usually restricted to the analysis in terms of entropic functionals in position space. Such is the case of a pioneering work36 regarding the information-theoretical analysis of interelectronic correlation in atomic systems, by considering a N-body density, defined from the position-space wave function Ψ(~r1, . . . ,~rN) of the N-electron atom. Then, the functionals Fisher information I, Shannon entroy power J in terms of the Shannon entropy S, the information product P= (IJ)/3 and the associated information plane I−Jwere considered, particularizing to the reduced oneand two-body densities. Other points, considered there, are: (i) the discussion of the main analytical properties regarding the aforementioned quantites (highlighting the superadditivity of I, the subadditivity of S, and the lower bound to the product P), (ii) a numerical analysis of those quantities, limited to the 6 helium-like (N= 2 electrons) systems with nuclear charge Z= 1 −5,10, (iii) the interpretation of the results, attending to the inter3 electronic correlation. More recent studies afford: (i) the analysis of the four helium-like systems H−, He, Li+ and Ps−(positronium negative ion)37, attending to the dependence of the position-space Shannon entropy on the continuous variable Z(nuclear charge), particularly around the critical value for which the system becomes unstable, and posing as future work determining the momentum space entropy and checking the uncertainty relation for the total entropy Sr+Sp≥3(1 + ln π); (ii) the analysis of three Rydberg series of He doubly excited states38, by means of Shannon entropy and Fisher information (for the reduced one-particle spatial density) and von Neumann and linear entropies (as measures of entanglement, from the reduced one-particle density matrix); and (iii) a comparative study39 of the atomic R´enyi entropies for the exponential-cosine screened Coulomb potential, by using different wave functions for the 1s2-state of the helium isoelectronic series. We want to complete those studies with a more extensive list of informational quantities, using these past results as a supporting floor, and extending them to all the neutral atomic systems in the Periodic Table. This paper is structured as follows: in the first section we show the formulation of the pair densities and how their aspect would be when used in the context of the Hartree-Fock method. The applied measures comprise the Shannon entropy, the disequilibrium and the LMC complexity, as well as the corresponding information planes. In the second section we will provide and discuss the numerical results regarding the measures showed in the previous section. Finally, some conclusions will be discussed and future works will be proposed 2ELECTRON PAIR DENSITIES AND RELATED INFORMATION THEORETIC MEASURES In terms of the N-electron wave function, the two-electron densities are defined as Γ(~r1,~r2) = ZΨ(~x1, ~x2,··· , ~xN)Ψ∗(~x1, ~x2,··· , ~xN)dσ1dσ2d~x3···d~xN(1) 4 in the position space, and Π(~p1, ~p2) = ZΦ(~y1, ~y2,··· , ~yN)Φ∗(~y1, ~y2,··· , ~yN)dσ1dσ2d~y3···d~yN(2) in the momentum space. The variables ~xi=~riσiand ~yi=~piσiare combined coordinates which include the spin. It is well known that the physical meaning of these densities regards the probability of finding an electron with given quantum numbers within the region ~r1d~r1if there is another electron with allowed/compatible quantum numbers within the region ~r2d~r2, and similarly regarding the momentum regions ~p1d~p1and ~p2d~p2. These densities are directly related to the electron correlations, as the compatibility of an electron state is organically determined by the compatibility of its state with those of the others. They naturally give us quantifiers of correlation between electrons. The two-electron densities are going to be calculated by the Hartree-Fock approach and can be expressed as follows40: Γ(~r1,~r2) = 1 N−1[Nρ(~r1)ρ(~r2)−Γx(~r1,~r2)] (3) in the position space, and Π(~p1, ~p2) = 1 N−1[Nγ(~p1)γ(~p2)−Πx(~p1, ~p2)] (4) in the momentum space, respectively. The functions ρ(~ri) and γ(~pi) are the one-electron densities and Γx(~r1,~r2) and Πx(~p1, ~p2) are the exchange densities in the position and momentum space respectively. Let us remark that, using Hartree-Fock functions, we are studying the Fermi correlation between same-spin electrons which arises from the antisymmetry of the wave function. In this sense, the term electron correlation, mentioned before, alludes to the statistical correlation. This point must be clarified in order to not lead to confusion if one considers a correlated system as one beyond the Hartree-Fock approximation (the L¨owdin definition of correlation energy). The Shannon entropy, S, of the normalized-to-unity electron pair densities are given by: S(Γ) = −ZΓ(~r1,~r2) ln Γ(~r1,~r2)d~r1d~r2(5) for the position space, and S(Π) = −ZΠ(~p1, ~p2) ln Π(~p1, ~p2)d~p1d~p2(6) 5 for the momentum space, respectively. This quantity measures the extent to which the density is spread, so that it is a measure of delocalization. Let us point out that the Shannon entropy could reach negative values. To avoid this and to guarantee that the uncertainty is non-negative, sometimes it is useful to define the exponential Shannon entropy as L=eS(7) Notice, in addition, that the exponential Shannon entropy is defined in this way in order to have the same dimensions as the variable considered, as the variance does, one of the most commonly used uncertainty measures. Considering that the electron pair density is, in its spherically averaged form, bidimensional, while its corresponding monoelectronic counterpart is monodimensional, a higher value on the Shannon entropy can be expected just due to the more natural spreading of the electron pair density. The Shannon entropy has already been employed in the past for studying electron pair densities10,20,21,26 showing how correlation effects can be successfully detected with this measure, and establishing relationships between its values and atomic properties. The disequilibrium, self-similarity41 or information energy42,D, quantifies the departure from uniformity of the probability density (equiprobability). In position space, the disequilibrium for two-electron density is given by D(Γ) = ZΓ2(~r1,~r2)d~r1d~r2,(8) and in momentum space is defined as D(Π) = ZΠ2(~p1, ~p2)d~p1d~p2.(9) . It is worthy to point out that both quantities Sand Dposses a global character, i.e., they consider the behavior of the distribution over its whole domain. Aside of the properties of the entropic measures described above, it is interesting to quantify the complexity of the physical systems. The characterization of complexity is not unique and the utility of each definition depends on the type of system or process, the 6 level of the description, and the scale of the interactions among the constituents of the systems considered, e.g., elementary particles, atoms, molecules, biological systems, etc. Fundamental concepts such as uncertainty or randomness are frequently employed in the definitions of complexity, although some other concepts such as clustering, order, localization or organization might be also important for characterizing the complexity of systems or processes. Here, we focus our attention on a complexity measure defined as a product of two information theoretical measures in order to simultaneously quantify two facets of the electron density of the system; namely, the LMC shape complexity,C(LMC). This quantity was introduced in 1995 by L´opez-Ruiz, Mancini and Calbet43 although, later on, it has been criticized44, modified45,46 and generalized47 leading to a useful estimator which reaches minimal values for both extremely ordered and disordered limits (i.e., for the Dirac-delta distribution and for the highly flat ones, respectively), satisfying also the desirable properties of invariance under scaling transformation, translation and replication48,49. The utility of this improved complexity has been clearly shown in many different fields50–52 allowing reliable detection of periodic, quasiperiodic, linear stochastic and chaotic dynamics43,48,49. The LMC complexity is defined by the product of two single-facet entropy measures (the disequilibrium Dand the exponential Shannon entropy eS) as CLMC (Γ) = D(Γ) ×eS(Γ) (10) in position space, and CLMC (Π) = D(Π) ×eS(Π) (11) in momentum space. This composite information-theoretic quantity measures the complexity of the system by means of a combined balance of the average height of the probability density (as given by D) and its total bulk extent (as given by S), i.e., the uniformity and delocalization features. This quantity satisfies the bound CLMC ≥1 for any probabilty density, with domain of arbitrary dimensionality53. 7 3RESULTS AND DISCUSSION In recent years, there has been an increasing interest in the information-theoretical analysis of interelectronic correlation in atomic systems. Different works have been performed within this field36–39, nevertheless, all of them deal with two-electron systems, usually restricted to the analysis in terms of entropic functionals in position space. This section is aimed to perform a comparative study among atomic oneand two-particle densities, on the basis of the respective information-theoretic functionals considered in the previous section. The analysis is two-fold, by considering densities in both conjugated spaces: that is, the position-space ones ρ(~r) and Γ(~r1,~r2), and the momentum-space ones γ(~p) and Π(~p1, ~p2). Functions normalized to unity will be managed in what follows, for the sake of their interpretation as probability distributions. Note that for the one-particle density, the Shannon entropy is given by S(ρ) = −Zρ(~r) ln ρ(~r)d~r, (12) S(γ) = −Zγ(~p) ln γ(~p)d~p, (13) in position space and momentum space, respectively. The disequilibrium for the one-electron density in position space can be defined as D(ρ) = Zρ2(~r)d~r, (14) and in momentum space is given by D(γ) = Zγ2(~p)d~p. (15) The LMC complesity for the one-electron density can be defined as CLMC(ρ) = D(ρ)×eS(ρ),(16) in position space, and CLMC (γ) = D(γ)×eS(γ)(17) in momentum space. 8 4CONCLUSIONS In this work, a variety of informational measures has been employed in order to perform a comparative study among atomic oneand two-particle densities. The aim of this analysis is to better understand the differences between these densities. The information measures employed for this purpose comprise Shannon entropy, disequilibrium and the LMC complexity measure. The Shannon entropy has been calculated for the one-electron and electron-pair densities in both position and momentum spaces. In position space we found a high resemblance between both quantities, with a similar extrema structure but a higher value for the twoparticle densities. As expected, a higher uncertainty for the electron-pair density as compared to the one-electron ones is confirmed. The analysis of their extrema structure has been carried out in detail. Thus, the Shannon entropy has proved its success when quantifying electronic configuration properties when applied to the electron-pair densities, providing even more sensibility to these qualities due to its higher value when compared with its one-electron alternative. It is worth mentioning that the resulting curves overlap, S(Γ) ∼2S(ρ), so that the electron spatial locations are roughly independent, at least in the atomic case. In momentum space, the resemblance was considerable as well, although in both oneand two-electron levels there is bare structure, just a monotonous increase. In this space, the overlap between both curves also occurs. We can say that the monoelectronic and electron pair Shannon entropies behave very similarly in position and momentum spaces and that, in terms of spreading and delocalization, both oneand two-electron densities behave in an analogous manner in both position and momentum spaces. The disequilibrium has been quantified in a comparable way for both kind of densities in both conjugated spaces. In position space, it showed an structure-free monotonous type of behaviour, much akin to Shannon entropy in momentum space. Even so, a relevant difference could be observed: the curves for one-electron and electron-pair densities intersect around Z= 10. For very light systems, the disequilibrium for the two-particle density is below that of the one-electron density. This is due to the low amount of electron pairs which reduces the effect of contraction provoked by the nucleus. This occurrence, along with the knowledge that 15 disequilibrium quantifies the relative strength of the nuclear attraction towards the electron cloud, is a measure of the exact point where the electron interaction significantly reduces the effect of the nuclear contraction. In momentum space, the disequilibrium displays identical structure of extrema, and roughly the same monotonic behavior as the Shannon entropy in position space. The LMC complexity in position space showed the same structure as the Shannon entropy, i.e., the same extrema structure, which means that the exponential entropy is the dominant factor for the LMC complexity. The spreading effect affects the electron-pair densities in a more relevant way as the nuclear contraction. Considering the electronic interaction mentioned above and how it affected the disequilibrium, the Shannon-dominated LMC is another proof of the more predominant effect of the electron interaction in the two-electron density when compared with the one-electron density. This change to the opposite situation in the momentum space, where the disequilibrium causes the appearance of extrema and Shannon entropy just modulates the structure. We can conclude that information theoretical measures have been proved successful when employed to quantify electron-pair densities characteristics. It has been proved the wellknown existing dissimilarities between one-electron and two-electron densities. Some previous studies, performed by other authors21,23, have been extended to new systems, confirming known tendencies and finding new ones. Research lines for future work include the definition of information measures based on the exchange densities, Eqs. (3) and (4), or the analysis of direct comparative functionals between the monoelectronic and the electron pair densities. So, it is expected to go beyond the actual results, based on the comparison of the results provided by each funtional at both the oneand two-electron levels, namely F(ρ) and F(Γ). Instead, one could consider quantifiers F(Γ, ρ) of dissimilarity or divergence which, in fact, would quantify electron correlation or mutual information in case of being non-negative, and vanishing only for independent variables. 16 ACKNOWLEDGMENTS This work was supported in part by the Spanish MINECO project FIS2014-59311-P (cofinanced by FEDER), and the grants FQM-4643 and FQM-7276 of Junta de Andaluc´ıa. A.L.M., J.C.A. and J.A. belong to the Andalusian research group FQM-020, and S.L.R. to FQM-239. 17 References 1. P. W. Ayers and M. Levy, J. Chem. Science 117, 507 (2005). 2. R. A. Gatenby and B. R. Frieden, Bull. Math. Bio. 69, 635 (2007). 3. R. O. Esquivel, J. C. Angulo, J. S. 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Figure 4: Disequilibrium for monoelectronic and electron pair densities, D(ρ) and D(Γ) respectively, in position space for atomic system with Z= 2 −103. Atomic units (a.u.) are used. Figure 5: Disequilibrium for monoelectronic and electron pair densities, D(γ) and D(Π) respectively, in momentum space for atomic system with Z= 2 −103. Atomic units (a.u.) are used. Figure 6: LMC complexity (CLMC) of the one-electron and the electron-pair densities, in (a) position and (b) momentum spaces, for systems with nuclear charge Z= 2 −103, in logarithmic scale. Figure 7: Disequilibrium-Shannon information plane, (a) D−Lplane for monoelectronic and electron pair densities, (b) D2−L2plane for monoelectronic densities and D−Lfor electron pair densities. Atomic units (a.u.) are used. 22 0.1 1 10 0 20 40 60 80 100 Shannon entropy Z One-electron density Two-electron density Figure 1a S. L´opez-Rosa, A.L. Mart´ın, J. Antol´ın, J.C. Angulo Int. J. Quant. Chem. 23 1 10 0 20 40 60 80 100 Shannon entropy Z 2*(One-electron density) Two-electron density Figure 1b S. L´opez-Rosa, A.L. Mart´ın, J. Antol´ın, J.C. Angulo Int. J. Quant. Chem. 24 1 10 100 1000 10000 20 40 60 80 100 LMC complexity Z One-electron density Two-electron density Figure 6b S. L´opez-Rosa, A.L. Mart´ın, J. Antol´ın, J.C. Angulo Int. J. Quant. Chem. 31 0.001 0.01 0.1 1 10 100 1000 10000 0.01 0.1 1 10 100 1000 Exponential Shannon entropy Disequilibrium One-electron density Two-electron density LMC complexity = 1 Figure 7a S. L´opez-Rosa, A.L. Mart´ın, J. Antol´ın, J.C. Angulo Int. J. Quant. Chem. 32 0.001 0.01 0.1 1 10 100 1000 10000 0.001 0.01 0.1 1 10 100 1000 10000 Exponential Shannon entropy Disequilibrium (One-electron density)2 Two-electron density LMC complexity = 1 Figure 7b S. L´opez-Rosa, A.L. Mart´ın, J. Antol´ın, J.C. Angulo Int. J. Quant. Chem. 33 Maxima Minima S(ρ)S(Γ)S(ρ)S(Γ) 3 4 13 13 10 10 20 20 18 18 25 25 24 24 31 32 30 29 38 38 36 36 43 42 50 50 46 46 57 56 54 54 84 84 78 79 89 88 86 86 Table 1: Local extrema of position-space oneand two-particle Shannon entropies, S(ρ) and S(Γ) respectively, for neutral atoms with nuclear charge Z= 2 −103. Atomic units (a.u.) are used. 34 Maxima Valence System subshell 3 Li 2s1 4 Be 2s2 13 Al 3p1 20 Ca 4s2 25 Mn 3d5 31 Ga 4p1 32 Ge 4p2 38 Sr 5s2 43 Tc 4d5 50 Sn 5p2 56 Ba 6s2 57 La 6s25d1 84 Po 6p4 88 Ra 7s2 89 Ac 7s26d1 Minima Valence System subshell 10 Ne 2p6 18 Ar 3p6 24 Cr 4s13d5 29 Cu 4s13d10 30 Zn 4s23d10 36 Kr 4p6 42 Mo 5s14d5 46 Pd 5s04d10 54 Xe 5p6 78 Pt 6s15d9 79 Au 6s15d10 86 Rn 6p6 Table 2: Valence subshell of systems corresponding to the local extrema of Shannon entropy in position space. 35