Analysis of correlation and ionization from pair distributions in many-electron systems

Eur. Phys. J. Plus (2021) 136:763 https://doi.org/10.1140/epjp/s13360-021-01747-8 Regular Article Analysis of correlation and ionization from pair distributions in many-electron systems S. López-Rosa1,2,a,J.C.Angulo 2,3,b, A. L. Martín4, J. Antolín2 1Departamento de Física Aplicada II, Universidad de Sevilla, 41012 Sevilla, Spain 2Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain 3Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, 18071 Granada, Spain 4BerlinInstituteforMedicalSystemsBiology,MaxDelbrückCenterforMolecularMedicineintheHelmholtz Association, 10115 Berlin, Germany Received: 21 January 2021 / Accepted: 10 July 2021 © The Author(s) 2021 Abstract Jensen–Shannon divergence is used to quantify the discrepancy between the Hartree–Fock pair density and the product of its marginals for different N-electron systems, enclosing neutral atoms (with nuclear charge Z=N) and singly-charged ions (N=Z±1). This divergence measure is applied to determine the interelectronic correlation in atomic systems. A thorough study was carried out, by considering (i) both position and momentum conjugated spaces, and (ii) systems with a nuclear charge as far as Z=103. The correlation among electrons was measured by comparing, for an arbitrary system, the double-variable electron-pair density with the product of the respective one-particle densities. A detailed analysis throughout the Periodic Table highlights the relevance not only of weightiness for the systems considered, but also of their shell structure. Besides, comparative computations between two-electron densities of different atomic systems (neutrals, cations, anions) quantify their dissimilarities, patently governed by shell-filling patterns throughout the Periodic Table. 1 Introduction The two-electron density Γ(r1,r2)is the probability density of any two electrons that are located at radii r1and r2, respectively, and is a convenient starting point to study the spatial interaction between two electrons in an explicit manner [1]. This density provides a quantummechanical description of the distribution of electron pairs in the space and therefore represents a central tool, in quantum chemistry, to study chemical binding [2]orelectronic structure and correlation in molecular systems [3,4]. On the other hand, the exact interaction energy of a many-electron system is determined by the electron pair density, but is not well-approximated in standard Kohn–Sham density functional models [5,6]. Consequently, pair density functional theory is regarded as not only an extension of the density functional theory but a reduced density matrix theory as well [7,8]. The main feature of the pair density functional theory is that the pair density, as a basic ae-mail: [email protected] (corresponding author) be-mail: [email protected] 0123456789().: V,-vol 123 763 Page 2 of 16 Eur. Phys. J. Plus (2021) 136:763 variable, essentially contains more information on the electron-electron interaction than the electron density [9,10]. The expectation value of an arbitrary two-particle operator can be obtained using only the electron pair density. In addition to this merit, the pair density functional theory has another, which is related to the expansion of the approximate functional. That is to say, in the pair density functional theory, the only required selection is the approximate form of the kinetic energy functional, because the exchange-correlation energy functional can rigorously be expressed by means of the pair density [11]. As a recent example multiconfiguration pair density functional theory has been developed, as a way to combine the advantages of both wave function and density functional theories to provide a better treatment of strongly correlated systems, and its results have been compared to those of some Kohn–Sham density functionals, including both traditional and modern functionals [12,13]. Additional applications have been performed recently with a diversity of molecular systems, on the basis of their oneand two-electron RDMs (reduced density matrices), which comparison provides measures of electron-correlation and entanglement among systems. These measures display their sensitivity to the presence of electric fields [14]. Therefore, given the above, it would be logical to assume that two-electron densities for atoms and molecules have been studied both intensively and extensively. However, this is not the case. The effort invested in interpretative studies of electron pair densities has been small as compared to that expended on such studies for one-electron densities. Indeed, there is a large amount of information coded inside monoelectronic densities; however, these densities do not give us any hint on how the position (or the momentum) of an electron conditions the positions (or the momenta) of the others. Besides it is worthy to note that within the Waller– Hartree theory, the two-electron density is related to the total X-ray scattering intensity which is accessible by experimental measurements [15,16]. One of the main difficulties rests in the interpretation of the electron pair density. Visual representation is a powerful analytical tool; however, being a function of 6 coordinates, graphical representation of the pair density is not feasible and manipulations to the twoelectron density must be carried out to extract useful information. One such manipulation gives place to the intracule and extracule densities and many relevant results have been obtained in this framework [17–21]. It is important to note that these densities consider the pairs of electrons as a whole thing. Other simplification consists in dealing with spherically averaged electron pair densities. Consequently, as a result of this reduction in dimensionality, some two-electron information is lost in the transformation. The concepts of uncertainty, randomness, disorder and delocalization are basic ingredients in the study, within an information-theoretic framework, of relevant structural properties for many different probability distributions appearing as descriptors of several chemical and physical systems and processes. The relevancy of these concepts has motivated new studies that pursue quantification, giving rise to a variety of density functionals, such as Shannon entropy [22], Fisher information [23], disequilibrium [24], complexity [25], and many others. These information measures have been widely employed for describing the information content and behavior in a great variety of fields [26,27] and, in particular, for the study of many-electron systems [27–34] Following the usual procedures carried out within information theory for quantifying the uncertainty or disorder of individual distributions, some extensions have been made in order to introduce the concepts of ‘distance’ and ‘divergence’ between two distributions, as comparative measures of their dissimilarity. Jensen–Shannon divergence (JSD) is a powerful divergence measure that represents the difference between the Shannon entropy of the mean density and the mean value of the indi123 Eur. Phys. J. Plus (2021) 136:763 Page 3 of 16 763 vidual entropies [35–37]. Its non-negativity arises from the convex character of the Shannon entropy functional. This comparative measure has found deep applications in statistics and many other fields, including entanglement characterization [38], analysis of symbolic sequences or series, and in particular in the study of segmentation of DNA sequences [39]. JSD and some effective generalizations have been used for the study and comparison of one-electron densities [40–44]. For the first time, to the best of our knowledge, this information measure of dissimilarity is applied to the study of two-electron density functions. Let us emphasize, however, recent applications of strongly related functionals (e.g. mutual information, cumulative residual entropies, interaction information) for uncoupled and interacting harmonic oscillators [45,46]. Additional applications include the cumulant part of two-electron RDMs, for the detection of van der Waals interactions between specific parts of molecular systems [47]. Other measures enclosing also the gradient of the density, as the relative Fisher information, were considered for the study of specific one-particle quantum systems (Morse potential, isotropic oscillator or hydrogen-like atoms) [48,49]. In this paper, small correlation effects in atoms are analyzed, on the basis of discrepancies between the Hartree–Fock oneand two-electron densities. In doing so, comparative density functionals are employed as quantifiers of dissimilarity among the above densities. So differences arising from the exchange term are emphasized, as the main cause of them. A detailed analysis throughout the Periodic Table reveals the relevance not only of weightiness for the systems considered, but also of their shell structure. Besides, comparative computations between two-electron densities of different atomic systems (neutrals, cations, anions) quantify the important differences patently governed by shell-filling patterns throughout the Periodic Table. Thereby, we use an information-theoretic functional that has shown its universality and versatility in other computations: the Jensen–Shannon divergence (JSD) [35–37]. The organization of the paper is the following: in the next section, two-particle densities are defined and presented for atomic systems and also the entropic divergence measure (JSD) we are going to use for comparing them. Numerical computations and results are presented in Sect. 3. Finally, conclusions and open problems or future work are briefly described in the last section. 2 Electron pair densities and related measures In terms of the N-electron wave function Ψ, and its Fourier transform Φ, the two-electron densities are defined as Γ(r1,r2)=Ψ(x1,x2,...,xN)Ψ ∗(x1,x2,...,xN)dσ1dσ2dx3...dxN(1) in the position space, and Π(p1,p2)=Φ(y1,y2,...,yN)Φ∗(y1,y2,...,yN)dσ1dσ2dy3...dyN(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 a compatible state within the region r1dr1if there is another electron with allowed/compatible quantum numbers within the region r2dr2,and similarly regarding the momentum regions p1dp1and p2dp2. These densities are directly related to the electron correlations, as the compatibility of an electron state is in organically 123 763 Page 4 of 16 Eur. Phys. J. Plus (2021) 136:763 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 follows [50]: 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 densitiesandΓx(r1,r2)andΠx(p1,p2)aretheexchangedensities 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 Löwdin definition of correlation energy). It should be noticed that the one-particle densities are marginal distributions of the twoparticle ones, because the former are obtained from the latter by integration, as ρ(r1)= Γ(r1,r2)dr2and γ(p1)=Π(p1,p2)dp2.Thus,theinformationcontentofthemarginals is not enough to determine the double-variable distribution they are coming from. Different definitions of distance between two arbitrary distributions f(x)and g(x)appear, where both are defined over the same domain. The interest in finding appropriate tools for ‘comparing’ two different systems or processes in terms of distribution functions is justified by the strong connection between some of the above mentioned information measures and many relevant physical and chemical properties of those systems. The term ‘distance’ should be understood as a measure of how dissimilar the two functions are, not necessarily verifying all mathematical properties required for a true distance, as the usual quadratic distance does. However, all of them keep as main distance properties the non-negativity, the symmetry (invarianceunderexchangeof functions) and saturation (minimal zero valueonly for identical distributions). The relative entropy or Kullback–Leibler divergence (KL) [35] is one of the pioneering global measures of the difference between two arbitrary unior multivariate probability distributions. It expresses the amount of information supplied by the data for discriminating among the distributions, being a ‘directed divergence’ and therefore not symmetric: KL(f,g)=f(x)ln f(x) g(x)dx(5) Its applications for different procedures in obtaining minimum cross entropy estimations and the determination of atomic and molecular properties [29], among others, make it to constitute an essential tool within the information theory. Won and You introduced, somewhat implicity, a closely related information measure between two or more distributions, the Jensen–Shannon divergence (JSD) [51,52]: JSD(f,g)=1 2KLf,f+g 2+KLg,f+g 2.(6) Consequently, JSD represents the mean dissimilarity (understood in terms of the KL measure) of each density respect to the mean one. 123 Eur. Phys. J. Plus (2021) 136:763 Page 5 of 16 763 This measure is strongly related to the Shannon entropy, which is defined for an arbitrary unior multivariate probability distribution as S(f)=−dxf(x)ln f(x). Shannon entropy is well known to play a relevant role within an information-theoretic framework because it constitutes a measure of spreading of the density over its domain [53]. Attending to the JSD definition given above, the Jensen–Shannon divergence can be also expressed in terms of the Shannon entropy as JSD(f,g)=Sf+g 2−1 2[S(f)+S(g)].(7) The JSD is characterized for quantifying the “Shannon entropy excess” of a couple of distributions with respect to the mixture of their respective entropies (mean density), defined in Eq. (7) as a particular case of the so-called weighted generalized divergences [54] with identical weights 1/2. Apart from preserving the global character of the Shannon entropy, the JSDpossessesthe main propertiesrequired for a measuretobe interpreted asan informational distance. In particular, its non-negativity is a consequence of the convexity of the S(f) functional. Other important advantages of this divergence are that: (i) does not require the condition of absolute continuity for the probability distributions involved, (ii) weights of each density can be different from 1/2 as appearing in Eq. (7), and (iii) can be generalized for an arbitrary number of distributions. During the past years researchers have been interested in work towards parametric generalizations of these classical measures of information. The JSD also admits other kind of generalizations, as will be shown elsewhere. In this work we intend to perform an exhaustive analysis of atomic systems using JSD, on the basis of their two-electron densities, in the following two ways: (i) Differences between the pair distribution Γ(r1,r2)and the product of marginals ρ(r1)ρ(r2)reveal the existence of some interconnection between its variables and quantify the effects from the exchange density on the systems under consideration. With JSD those differences are obtained for atomic systems (neutrals and ions), in both conjugated spaces (Sect. 3.1). (ii) The ionization of a neutral system, by adding or removing an electron, modifies its structural characteristics. This was studied by means of the JSD between the one-particle distributions of the neutral and the ionized systems [55]. Here, a step further is taken, dealing with divergences among electron-pair densities, also in position and momentum spaces (Sect. 3.2). 3 Numerical results Divergence measures allow us to quantify the differences between general systems, in particular by means of electron atomic densities. In this section we intend to employ Jensen– Shannon divergence in order to analyse the particularities and characteristics of two-electron and one-electron densities. The Near-Hartree–Fock wavefunctions of Koga et al [56,57]are employed. Functions normalized to unity will be managed in what follows, for the sake of their interpretation as probability distributions. 3.1 Interelectronic correlation analysis We will compute the Jensen–Shannon divergence between two-electron density and the product of one-electron densities, being one of the main reasons its property that its value is zero when the systems compared have no correlation. Thus, the Jensen–Shannon divergence 123 763 Page 6 of 16 Eur. Phys. J. Plus (2021) 136:763 Fig. 1 Jensen–Shannon divergence in position space (JSDr) between the electron pair density and the product of monoelectronic densities, for neutral atoms with number of electrons N=3−103. Labels are for systems with closed shells (red), closed subshells (blue) or anomalous shell-filling (green). Atomic units (a.u.) are used actsasameasureofthedisparitybetweentwo-andone-electrondensities.Thesecomparisons will be carried out on double variable distributions in both position (Γ(r1,r2)and ρ(r1)ρ(r2) respectively)andmomentum (Π(p1,p2)andγ(p1)γ (p2)respectively)spaces.Let us remark that in this case, the last term in the JSD (see Eq. (7)) can be expressed as −S(Γ) 2+S(ρ)in position space, and similarly for the momentum space. The systems under study are neutral atoms with a number of electrons N≤103 (nuclear charge Z=N), and singly-charged ions (Z=N±1) with N≤54. Firstly,wecomputedtheJensen–Shannondivergencebetweentheelectronpairdensityand the product of the corresponding monoelectronic densities for neutral systems N=2−103 in position space. The obtained results are shown in Fig. 1. It can be observed the decreasing trend of JSD when the nuclear charge of the system increases, which indicates that the correlation decreases as the number of electrons increases. This was an expected result, as the number of electrons is known to determine how strong or weak is the relevance of the correlation between the mentioned electrons. Thus, in the case that we have information about the position of an electron within a group of three, this knowledge becomes much more relevant than if we have a group of thirty electrons, when the information about just one does not tell us much more about the other twenty nine. Notwithstanding, some additional patterns are noticeable, most certainly related to the shell filling structure. This fact was emphasized in a previous work [58] for systems within the shorter range N≤36, on the basis of the mutual information as quantified by the KL divergence among distributions, as here done by means of JSD up to N=103. The trend is monotonous, without many exceptions for a long period, however some discontinuities appear when changing from a period to another indicating a transition to a new electronic shell. Each segment can be assigned to a particular shell, in the fashion the figure shows. Besides, some less relevant extrema appear, all of them related to anomalous shell filling systems. Some specific systems have been labelled, as follows: – Closedshells: N=10,18,36,54,86.Thusnoblegasesappearasminimasystematically. The same was observed in the KL divergence for systems up to N≤36 [58]. 123 Eur. Phys. J. Plus (2021) 136:763 Page 7 of 16 763 – (i) Anomalous shell-filling: minima N=42,46 (outer d-subshell), with valence subshell half-filled and filled, respectively. (ii) The same applies to N=24,29: in spite of not being extrema, the curve displays a ‘change of slope’ at those points, to be contrasted later with momentum space. (iii) Minor extrema appear for heavy systems, within a region of anomalies. – Closed subshells: minima N=48,80 with filled d-subshell. The Jensen–Shannon divergence between electron pair density and the product of monoelectronic densities in momentum space hardly provides additional information. The conclusions we arrive are similar than before, in fact, the curves in both spaces almost overlap each other, as it can be seen by comparing Fig. 2(momentum space) with the previous one (note the identical scales of both figures). Thus the electron pair densities in momentum space are providing roughly the same information as the position ones. Nevertheless, additional remarks are in order. Comparing curves in position and momentum space, it is observed that (i) the aforementioned ‘changes of slope’ at N=24,29 in position space are displayed as maxima in momentum space; (ii) previous minimum N=42 translates now to N=43, also with outer d-subshell; (iii) instead of minimum N=80 it is now observed the maximum N=78. No extrema appear in the heavy systems region. Notice that just mentioned systems N=24,29,42,43,46,78 have a non-filled inner ssubshell and an outermost d-subshell (the inner one is empty for the system N=46, clearly highlighted in both figures). As observed in Ref. [58] for KL, most JSD values are higher in position than in momentum space. The opposite applies for a few systems with anomalous shell-filling. Particularly relevant are the cases N=24,29 with half-filled 4sinner orbital and outermost 3dsubshell (half-filled and completely filled, respectively). Their JSDp(Fig. 2) are greater than the respective JSDr(Fig. 1) by 17% and 25%. Such amount belongs to the range 8−13% for heavier systems with similar anomalies: N=41,42,44,45,47 (subshells 5s4d), and N= 78,79 (subshells 6s5d), all them with an inner half-filled s-orbital. It is concluded that, for most systems, correlation effects are weaker in momentum space, the opposite being true for the aforementioned anomalous systems, in a slight fashion with few exceptions. These results support and extend the conclusions obtained in Ref. [58] for systems up to N=36, regarding the relative behaviors in position and momentum spaces. The above discussion was focused on neutral atoms and their properties, but in what follows we will consider also some of their corresponding ions. Let us now use the Jensen– Shannon divergence, JSD, in order to quantify the differences between the electron distributions at the oneand two-electron levels, now for ionized systems. In doing so, Fig. 3shows the JSD between the monoelectronic and the electron pair density of different ions, calculated in both conjugated spaces. First let us focus on Fig. 3a where singly-charged cations with number of electrons N=2−54 have been taken into account. The curves show us a decreasing behaviour when Nincreases and, in this case, the momentum space roughly presents smaller values of JSD. However, an interesting pattern appears in the figure: cations corresponding to N=10,18,23,28,36,41,46,48,54 have a minor difference, a lower value of JSD, in both conjugated spaces with the only exception N=48 (extremely light minimum, only in position space). Among maxima (located at almost identical positions in both spaces), let us emphasize the presence of N=24,29,42. Regarding these two collections, some comments are in order: – Asforneutral atoms, noble gases N=10,18,36,54are displayed as lowerpeaks.On the other hand, alkalines N=3,11,19,37 appear as maxima. Similarly occurs, in position 123 763 Page 8 of 16 Eur. Phys. J. Plus (2021) 136:763 Fig. 2 Jensen–Shannon divergence in momentum space (JSDp) between the electron pair density and the product of monoelectronic densities, for neutral atoms with number of electrons N=3−103. Labels are for systems with closed shells (red) or anomalous shell-filling (green). Atomic units (a.u.) are used space, with the closed-shell system N=48 (minimum) and its neighbors N=47,49 (maxima). – Pairs of extrema 23−24 and 28−29 (minimum–maximum) are due to the anomalous shell filling of both maxima N=24,29, with the inner half-filled 4ssubshell, and with half-filled or filled (respectively) outermost 3dsubshell. – Similarly the pair of consecutive extrema 41 −42 has inner half-filled 5ssubshell, and outermost subshell 4d, half-filled for the maximum N=42. – Particularly interesting is the strong minimum N=46, as discussed for neutrals, with empty 5ssubshell and filled valence subshell (4d). – A few of the above systems display a higher JSD in momentum than in position space: N=24,29,42,47 (with inner half-filled s-orbital and outermost half-filled or completely filled d-orbital), as well as N=25 (half-filled outermost d-orbital). In Fig. 3b, where the anions have been considered, exactly the same pattern is observed, but with a less elaborate structure, due to the fewer available systems as it was previously mentioned. Some alkaline earths have been labelled, but let us emphasize the ‘absence from the list’ of alkalines, which are displayed as maxima for cations. Position space JSD is almost systematically greater than the momentum one. The only extremely light exceptions are N=41−42, with differences between spaces below 0.2%. 3.2 Atomic ionization Applying divergence measures to the analysis of ions will allow us to find which electrons have a higher impact on the whole density. In order to do that, we inspect the Jensen–Shannon divergence between the pair densities of an ion (cation or anion) and its respective neutral, in both position and momentum spaces. For completeness, similar results on the basis of one-particle densities, previously obtained [55], are also displayed. This is shown in Figs. 4 and 5. Next results are to be interpreted as measures of the ‘amount of change’ on the electron cloud of a given system after its ionization takes place, either by ejecting or by capturing an 123 Eur. Phys. J. Plus (2021) 136:763 Page 9 of 16 763 (b) (a) Fig. 3 Jensen–Shannon divergence, in position and momentum spaces, between the electron pair density and the product of one-particle densities, for acations and banions with number of electrons 3 ≤N≤55. Labels are for systems with closed shells (red), closed subshells (blue) or anomalous shell-filling (green). Atomic units (a.u.) are used electron. Thus, if the initial neutral system has Nelectrons, the resulting ion will have N−1 or N+1 electrons, for the cation and the anion, respectively. This interpretation applies to both conjugated spaces, as well as to one-particle and electron pair densities. Figure 4encloses curves of JSD between the electron distribution of a neutral system and that of its singly-charged cation, denoted in position and momentum spaces as JSDr(N,C)in Fig. 4a, and JSDp(N,C)in Fig. 4b, respectively. A simple function of the atomic ionization potential AIP needed for performing such process is also depicted. This function has been considered in order to better compare all curves, which display a variety of local extrema, mostofthemcharacterized accordingly with the shell-fillingofneutrals and ions, as discussed below. 123 763 Page 16 of 16 Eur. Phys. J. Plus (2021) 136:763 43. A.L. Martín, S. López-Rosa, J.C. Angulo, J. Antolín, Jensen-Shannon and Kullback–Leibler divergences as quantifiers of relativistic effects in neutral atoms. Chem. Phys. Lett. 75, 642 (2015) 44. J. Antolín, J.C. Angulo, S. Mulas, S. López-Rosa, Relativistic global and local divergences in hydrogenic systems: a study in position and momentum spaces. Phys. Rev. A 90, 042511 (2014) 45. 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