Shannon entropy in confined He-like ions within a density functional formalism

Shannon entropy in position ($S_{\rvec}$) and momentum ($S_{\pvec}$) spaces, along with their sum ($S_t$) are presented for unit-normalized densities of He, Li$^+$ and Be$^{2+}$ ions, spatially confined at the center of an impenetrable spherical enclosure defined by a radius $r_c$. Both ground as well as some selected low-lying singly excited states, \emph{viz.,} 1sns (n $=$ 2-4) $^3$S, 1snp (n $=$ 2-3) $^3$P, 1s3d $^3$D are considered within a density functional methodology that makes use of a work-function-based exchange potential along with two correlation potentials (local Wigner-type parametrized functional as well as the more involved non-linear gradient- and Laplacian-dependent Lee-Yang-Parr functional). The radial Kohn-Sham (KS) equation is solved using an optimal spatial discretization scheme via the generalized pseudospectral (GPS) method. A detailed systematic analysis of the confined system (relative to corresponding free system) has been performed for these quantities with respect to $r_c$ in tabular and graphical forms, \emph{with and without} electron correlation. Due to compression, the pattern of entropy in aforementioned states gets characterized by various crossovers at intermediate and lower $r_c$ regions. The impact of electron correlation is more pronounced in weaker confinement limit, and appears to decay with rise in confinement strength. The exchange-only results are quite good to provide a decent qualitative discussion. The lower-bounds provided by entropic uncertainty relation holds good in all cases. Several other new interesting features are observed.


I. INTRODUCTION
A particle in an impenetrable box of infinite height has served the role of a simple, elegant pedagogical tool to illustrate the effects of boundary condition on energy spectrum of a quantum system. Understanding of such a system in some sub-region Ω of space (in contrast to "whole" space available in free system) offers new insights to simulate realistic situations in highly inhomogeneous media or in an external field. Matter constricted under such extreme pressure environment gives rise to a wide range of novel changes (from respective free counterpart) in energy spectra, electronic structure, chemical reactivity, ionization potential, polarizability etc., depending on geometrical forms of cavity and dimensions. This has inspired a variety of theoretical and experimental works. Some prominent applications are found in the context of cell model of liquid, superlattice structure, quantum dot, quantum wire, atoms/molecules encapsulated inside nanocavities (like fullerene, zeolite sieves, porous silicon, carbon nanotube), modelling defects in solids, confined phonons (or plasmons, polaritons, gas of bosons), as well as astrophysical phenomena such as mass-radius relation of white dwarfs, ionized plasma etc. The topic is vast and there has been a burgeoning growth of activity as evident from an extensive literature having many excellent comprehensive reviews. Interested reader may refer to following reviews [1][2][3][4][5] and references therein.
The first report of a confined hydrogen atom (CHA) within a sphere having rigid impenetrable walls was published as early as in 1937 [6] imposing the Dirichlet boundary condition that the wave function vanishes at boundary. Subsequently, many attempts have been made to estimate the eigenvalues and eigenfunctions invoking a wide range of approximate analytic, semi-analytic and purely numerical schemes. Here we mention a few ones like Rayleigh-Schrödinger perturbation theory, Wentzel-Kramers-Brillouin method, power-series solution, hypervirial theorem, Padé approximation, Lie algebraic treatment, super-symmetric quantum mechanics, Lagrange-mesh method, asymptotic iteration method, searching the zeros of hypergeometric function, generalized pseudospectral (GPS) method, Hartree-Fock (HF) method [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In recent years, exact solution of the Schrödinger equation has been found in terms of Kummer-M function (confluent hypergeometric) [16]. As the boundary approaches nucleus and volume of confinement squeezes, one notices a monotonic increase in energy in CHA. Another interesting feature is that, in contrast to free atom, due to breaking of symmetry in CHA, it is characterized by different energy eigenvalues, eigenfunctions and re-duced degeneracies. On the other hand, new degeneracies, namely, simultaneous, incidental and inter-dimensional degeneracy, which are non-existent in the free system, is introduced in CHA afresh. Apart from the effect of compression on ground and various energy levels, properties such as dipole shielding factor, nuclear magnetic screening constant, hyper-fine splitting constant, pressure, static and dynamic polarizability, etc., were examined.
All the above works pertain to the wave function-based methods. In the past two decades, some results have been published within the alternative density-based concept-the so-called density functional theory (DFT) [47][48][49]. Thus within an exchange-only framework (using two exchange functionals, viz., local density approximation (LDA) [47], and Becke-88 exchange potential [50]), the desired Kohn-Sham (KS) equation was solved satisfying the Dirichlet boundary condition for many-electron systems via numerical shooting method [51].
The usefulness of a one-parameter hybrid exchange functional (including a fraction of exact exchange and Perdew-Burke-Ernzerhof functional) for treatment of confined atoms, has been presented lately [52]. In another attempt, ground and 1s2s 3 S, 1 S states of confined He atom were reported [53] taking into account the LDA-approximated exchange-correlation (XC) (with Perdew-Wang parametrization for correlation [54]) with and without self-interaction correction. Response properties such as polarizability and hyperpolarizability of confined He atom were reported within a DFT-based variation-perturbation approach [55]. In a recent work [56], spherically confined atoms were treated by means of local exchange potential corresponding to Zhao-Morrison-Parr and Becke-Johnson potential. Moreover, spherical confinement was used in the comparative study (taking free-ion limit as reference) of behavior of spin potential and pairing energy of first row transition metal cations within KS model [57]. A detailed analysis of correlation energy, performance of several commonly used functionals, electron density as well as the XC potential in some constrained atoms, has been reported [58,59]. The calculation of static polarizability of confined He and Ne atoms was done through time-dependent DFT in [60].
Recently there has been a growing interest in information theoretical analysis of diverse model and realistic systems. They have found wide-spread applications in many branches of physics and chemistry, such as thermodynamics, spectroscopy, quantum mechanics. In chemical physics, typically they can provide valuable information regarding localizationdelocalization, diffusion of atomic orbital, periodic properties, spreading of electron density, correlation energy, etc. Entropic uncertainty measures based on these quantities are arguably the most effective quantifiers of uncertainty, as they do not relate to any specific points of the respective Hilbert space. The present work is particularly concerned with Shannon entropy (S) [61,62], which is the arithmetic mean of uncertainty. Interestingly S like some of the other measures such as, Fisher information, Onicescu energy and Rényi entropy are functionals of density, and also characterize density. Many articles have been published to analyze these measures in free systems (e.g., for free He, we refer the reader to a recent article [63]), but in confined quantum systems as treated here, analogous studies are quite limited. Two such reports [64,65] in CHA are available so far. A systematic variation of S with respect to r c , in r and p spaces has been presented only lately [66,67] for ℓ = 0 as well as non-zero-ℓ states. One finds that, confinement affects S more profoundly in the stronger regime. Further, S r increases with rise in r c and at very low-r c region (≈ 0.1), CHA displays exactly opposite trend from a free H atom (S r declines with rise in n keeping l fixed).
Usually, the effect of perturbation on higher quantum number states is more pronounced.
In confined two-electron isoelectronic series (H − , He, Li + , Be 2+ ), S has been reported for only ground states [64] by means of BLYP calculations.
Some reports [52,[68][69][70][71][72][73] are available on S in penetrable confinement in atoms. For example, it was observed [68] that, in CHA, up to a certain value of r c , S r decreases with r c .
However, for small r c and depending on barrier height, S r may also increase. Apart from constant potential, it was probed [73] for confinements imposed by a dielectric continuum and by isotropic harmonic potential. It was also proposed [72] as an indicator to measure the delocalization of electron density. Ground-state atomic S's, as function of width of confining potential was calculated by employing the correlated Hylleraas-type wave function in both repulsive and attractive finite potentials [70]. Confinement by an inert geometric planar boundary with finite barrier height has been studied [69] within a Thomas-Fermi-Dirac-Weizsäcker-type DFT framework. Some limited works exist on excited states as well, viz., low-lying singly [70] and doubly [71] excited states. It is worth noting that, S values in He are available [74] in selected excited states, such as 1,3 S e , 1,3 P o and 1,3 D e .
The objective of this work is to make a thorough systematic analysis of S in a Helike ion placed inside a spherical cage or radius r c . This is done by invoking DFT within a work-function-based exchange potential in conjunction with two correlation functionals, viz., a local, parametrized Wigner-type [75] and somewhat involved nonlinear Lee-Yang-Parr (LYP) [76] functional. The pertinent KS differential equation is solved within the Dirichlet boundary condition by means of GPS method in an accurate efficient manner. The electron density as well as S r is calculated from the self-consistent orbitals. The p-space orbitals are constructed from respective r-space orbitals via standard Fourier transform, from which the S p 's are computed. Variation of S r , S p and total Shannon entropy sum (S = S r + S p ) with respect to r c is offered for He, Li + , Be 2+ . Apart from ground state, we also consider singly excited 3 S, 3 P, 3 D states arising out of configurations 1sns (n = 2-4) 3 S, 1snp (n = 2-3) 3 P and 1snd (n = 3). As apparent from the preceding discussion, there is a lack of such results in literature, especially in excited states, and we attempt to provide them. The article is organized as follows. Section II outlines the methodology used, Sec. III discusses the results along with comparison with available references, while Sec. IV makes a few concluding remarks.

II. METHODOLOGY
Here we briefly outline the proposed density functional method for ground and excited states of an arbitrary atom centered inside an impenetrable spherical cavity, followed by the GPS method for calculation of eigenvalues and eigenenergies of corresponding KS equation.
It may be noted that the present method has been very successfully used for ground and various excited states (such as singly, doubly, triply excited states corresponding to low-and high-lying excitation, valence and core excitation, autoionizing states, hollow and doubly hollow states, very high-lying Rydberg states, satellites states etc.) of free or unconfined neutral atoms as well as positive and negative ions in a series of articles [77][78][79][80][81][82][83]. But it has never been tested for any confinement studies as intended here. Thus we present an extension of the method for the purpose of confinement effects. Our focus remains on essential portions, omitting the relevant details, which could be found in above references.
The starting point is the non-relativistic single-particle time-independent KS equation with imposed confinement, which can be conveniently written as (atomic unit employed unless otherwise mentioned), where H is the perturbed KS Hamiltonian, written as, In the above, v ne (r) and v conf (r) signify external electron-nuclear attraction and the effective confining potentials, whereas second and third terms in right-hand side denote classical Coulomb (Hartree) repulsion and many-body XC potentials respectively. The desired confinement effect is built into the system by introducing a potential of following form (r c refers to the radius of spherical enclosure), Though DFT has achieved impressive success in explaining the electronic structure and properties of many-electron system in ground state in past four decades, calculation of excited-state energies and densities has remained a bottleneck. This is mainly due to the absence of a Hohenberg-Kohn theorem parallel to ground state, as well as the lack of a suitable XC functional valid for a general excited state. In this work, we have employed an accurate work-function-based exchange potential, which is physically motivated [84,85].
Accordingly, exchange energy is interpreted as the interaction energy between an electron at r and its Fermi-Coulomb hole charge density ρ x (r, r ′ ) at r ′ , and given by, Assuming that a unique local exchange potential v x (r) exists for a given state, it can be defined as the work done in bringing an electron to the point r against the electric field generated by its Fermi-Coulomb hole density, leading to the following form, where the electric field is expressed as, The Fermi hole can be written in terms of orbitals as, where |γ(r, is the single-particle density matrix and ρ(r) is the electron density, expressed in terms of occupied atomic orbitals (n i denotes occupation number) as, While the exchange potential v x (r) corresponding to a given state arising from an electronic configuration can be accurately calculated by the above procedure as delineated, the correlation potential v c (r) is unknown and must be approximated for practical calculations.
The current work incorporates two correlation functionals, namely, a Wigner-type [75] and LYP [76]. These two functionals have been chosen on the basis of their success in the unconfined atomic excited states, which are recorded in the references [77][78][79][80][81][82][83]. This work will help shed some light on the applicability of such functionals in the context of confined quantum systems, including those studied here.
With this choice of v x (r) and v c (r), the resulting KS differential equation, needs to be solved, where v ef f (r) is as defined in Eq. (2), maintaining the Dirichlet boundary condition. For an accurate and efficient solution, we have adopted GPS scheme leading to a non-uniform, optimal spatial discretization. It is simple but very effective method; the success has been demonstrated for many static and dynamic properties of a variety of singular and non-singular potentials of physical and chemical interest [80][81][82][83][86][87][88][89] such as, Coulomb, Húlthen, Yukawa, logarithmic, spiked oscillator, Hellmann potential, etc., along with its recent extension to quantum confinement [15,21,22]. As the method is very well established and documented, in the following, we will mention a very brief summary of it; the details are available in the cited references.
The key characteristic of this approach is to approximate an exact function f (x) defined which ensures that the approximation be exact at the collocation points x j , Here we utilize the Legendre pseudo-spectral method where x 0 = −1, x N = 1, while x j (j = 1, ...., N − 1)'s are defined by roots of first derivative of Legendre polynomial P N (x), with respect to x, namely, In Eq. (10), g j (x) are termed cardinal functions, and as such, expressed as, fulfilling the unique property that, g j (x j ′ ) = δ j ′ ,j . Then use of a non-linear mapping followed by a symmetrization procedure, eventually leads to a symmetric eigenvalue problem, which is solved by standard available softwares to provide accurate eigenvalues and eigenfunctions.
The p-space wave function is obtained numerically from Fourier transform of respective r-space counterpart in the following way, It is to be noted here that ξ(p) is not normalized; hence needs to be normalized. The normalized r-and p-space densities are represented as ρ(r) = N i=1 n i |φ i (r)| 2 and Π(p) = N i=1 n i |ξ i (p)| 2 respectively, where n i represents the occupation number of each orbital.
Next S r , S p and Shannon entropy sum S t are defined in terms of expectation values of logarithmic probability density functions, which have the forms given below as, Here ρ(r) and Π(p) are both normalized to unity.
All the computations are done numerically. The convergence is ensured by carrying out calculations with respect to variation in grid parameters, such as total number of radial points and maximum range of grid. It is generally observed that convergence is achieved relatively easily in the lower r c region compared to the r c → ∞ limit. All results given in the following tables and plots have been checked for above convergence.

III. RESULT AND DISCUSSION
At the onset, it would be appropriate to mention a few points to facilitate the discussion.
The net information measures in r and p space of confined many-electron system consist of To start with,    this table further reinforces the previous conclusions [64,72] that impenetrable walls impose confinements in a way that localizes the electron density, and consequently S r → −∞ when r c → 0.
However, it is important to point out that, in all these cases total energy monotonically decreases with r c eventually reaching the free-atom limit. Actually, with reduction in r c the r-space electron density gets compressed and as a consequence, S r decreases. On the contrary, S p gradually abates with progress in r c . At all r c 's, however, S t maintains the lower bound (6.434) governed by the well-known BBM inequality [62]. The qualitative behavior of S r , S p , S t with growth in r c does not change much with atomic charge (Z), although their numerical values differ. In fact, at a given r c , S r regresses and S p progresses with advancement of Z. With rise in Z, electron density gets compressed and hence such a pattern is noticed. Interestingly, while in one-electron systems S t does not depend on Z, in a many-electron atom, with change of Z it varies [94]. Earlier S t has been mentioned as a measure of correlation in free systems. Our work establishes the same fact in confinement as well. It is noticed that, for all the three species, S's are identical at very low r c region (≤ 0.5), without or with (either Wigner or LYP) correlation. Furthermore, these two results begin to differ at larger r c indicating correlation effects to assume more significance in the respective free-atom case. In other words, this implies that, at smaller r c region, XC effect is minimum, which enhances with rise in r c . Similar conclusions have been found in the energy analysis of confined He in [55]. For He and Li + , these have been estimated by BLYP calculation [64] in most of the r c 's considered here, which are appropriately quoted. Note that the HF values [90] for S r , S p , S t in ground state of unconfined He match reasonably well with our X-only results. Since we are unable to find reference theoretical results for X-only S's in the hard confinement, for direct comparison, as a matter of check, a couple of comparisons on respective free systems is provided here. Thus the literature S r , S p , S t values of He and Li + within HF method [90,91], employing N-normalized densities, given in footnote of the table, are in reasonable agreement with our X-only values. Recently, in a penetrable confinement calculation within HF, some results on S r have been presented [72]. The same has also been calculated from a DFT-based study with hybrid exchange functional [52].
Results for He in free-limit from both these studies, presented in footnote show quite decent agreement with ours. It may be noted that in these two aforementioned references S r in U 0 → ∞ corresponds to the impenetrable confinement. Highly accurate benchmark-quality result for S r was calculated from a Hylleraas-type variational method producing a value of 2.7051028 and 1.2552726 for free He and Li + [92] respectively. Correlated results obtained from variational Monte carlo and diffusion Monte Carlo methods [91] for He and Li + are also cited in the footnote. The current single-determinantal approach quite nicely compares with reference values in the table-XC-LYP providing a slight edge over XC-Wigner. It would be worthwhile to make a comparative energy analysis of these two functionals, which we intend to do in near future. No reference entropies are available for Be 2+ .
A careful examination of Table I reveals that, X-only, XC-Wigner and XC-LYP results provide similar qualitative trend with respect to changes in r c , for all three species. Hence in Fig. 1, X-only S r , S p and S t are plotted for ground state of all three isoelectronic members, as functions of r c in panels (a)-(c). The first two panels imply that, for a fixed Z, S r , S p go up and down respectively with rise in r c . On the contrary, for a given r c , variation of these two quantities with Z shows opposite trend; the former decays and latter develops as Z advances. These results reinforce the inferences drawn from Table I. Another point to be noted here is that, with lowering in r c the difference between S r and S p corresponding to two successive members of the isoelectronic series, diminishes; in other words, as r c enhances, so does the difference. As r c declines, both average electron-nucleus and electron-electron distances fall down. In stronger confinement regime, the effect of Z on ground state gets dominated by confining potential, resulting in the fact that the three S r , S p plots very nearly coincide. For a fixed Z, variation of S t with r c in panel (c) suggests that the entropy sum reduces dramatically from its free atomic value as r c is lowered. With enhanced confinement, a distinct minimum followed by a maximum shows up in the curve for all He-like ions. The minimum tends to shift towards left as Z progresses. This observed pattern is in consonance with that found in [64].
Now we move on to some low-lying singly excited state-S r , S p , S t for He, Li + and Be 2+ in Tables II-IV. Following the presentation strategy of previous table, it reports results for 1s2s 3 S, 1s2p 3 P and 1s3d 3 D states respectively. Like the ground state, here also in all three states, both X-only and correlation-included S r , S p moves up and down respectively with growth in r c . In all cases again S t obeys the stipulated lower bound [62]. The qualitative pattern of S r , S p , S t with progress in r c remains invariant with change of Z. However, their numerical values alter substantially. Similar to the ground state, at a fixed r c , S r falls off and S p enhances with advancement of Z. Once again, for a given Z, at low r c region, X-only, XC-Wigner and XC-LYP results practically merge with each other, signifying that the effect of correlation is somewhat less impactful in stronger confinement region, as the confining potential leads the contribution in this scenario. Whereas, with rise in r c , the correlation effect prevails, indicating its importance in free conditions. Except the free atom-limit of S r in 1s2s 3 S He, no reference results could be found for any of the confined states and we hope the present work would provide useful guideline in future.
In order to gain a better understand of the effect of confinement on excited states, S r , S p in compressed He have been plotted in panels (a), (b) of Fig. 2, for three triplet singly excited states arising from configuration 1sns, corresponding to n = 2 − 4. Since correlation does not affect the results qualitatively, for this purpose, it suffices to consider X-only results.
With this in mind, here and in next figure, only X-only entropies are shown. It is obvious from these plots that, for excited states, as in Fig. 1, S r gains with rise in r c , while S p declines. Now for a fixed r c , the behavior of S r with n (in this series), shows interesting pattern. For a large enough value of r c , which corresponds to the free-atom limit of He of the state under consideration, S r progresses as n grows. Though it may not be so apparent from the data presented in respective table or plot, as the maximum range of r c presented here is 10 a.u. This can be concluded from the fact that S r for 1s2s, 1s3s and 1s4s triplet states in the free limit are 5.20, 6.53, 7.43 respectively, signifying a progressive delocalization.
But this pattern gets dissolved with reduction in r c and crossing between S r for different respectively. Hence, one encounters frequent change in the order arrangement on proceeding from free to strong confined regime. In   However, at r c → ∞ limit and r c = 0.1, S p shows opposite trend to that of S r .
As a continuation of the earlier discussion, we present in Fig. 3, X-only S r , S p and S t as a functions of r c corresponding to 3 S, 3 P and 3 D states resulting from 1s3s, 1s3p and 1s3d configurations of He, in panels (a)-(c). From panel (a), at r c ≈ 10, we obtain the following ordering of entropy: S r ( 3 S) > S r ( 3 P ) > S r ( 3 D), indicating a drop in fluctuation as one passes from 3 S to 3 P to 3 D. Similar to that in Fig. 2, here also multiple crossovers take place at intermediate and lower r c region and eventually settles with the following sequential order S r ( 3 P ) > S r ( 3 S) > S r ( 3 D) at (r c ≈ 0.1), representing strong confinement region. Now in conjugate space, at higher r c region ≈ 20, S p displays an exact opposite trend to that of S r in the free limit, which is depicted in panel (b). This is obvious as the more localized a state is in r space, the more diffused it is in p space. Here also due to crossover between states this pattern gets dissolved at lower r c , leading to an ordering as S p ( 3 S) > S r ( 3 P ) > S r ( 3 D), at r c ≈ 0.1. Panel (c) portrays the response of S t which verifies that the lower bound is maintained throughout entire confinement region.

IV. CONCLUSION
Shannon information entropy (in r and p spaces) has been analyzed for confined He iso-electronic series. Ground and excited states were studied via a simple DFT method, by solving the radial KS equation through a generalized Legendre pseudospectral method. Some attempts are known for S of free He atom, as well as its confinement within a soft, penetrable boundary. However to the best of our knowledge, this is the first such systematic study of information in confined two-electron atom within a rigid, impenetrable spherical cage. Apart from ground state, several low-lying singly excited triplet states of the iso-electronic series are considered. As the X-only entropies are comparable to their HF counterparts in the free-atom limit, it is expected that this will also hold in the confined case as well. The effects of electron correlation have been probed through two correlation functionals. For the states considered here, the correlation contribution remains rather small in low r c regime, assuming greater significance as the latter approaches free-atom limit. It is observed that the two correlation functionals offer quite comparable results as far as Shannon entropy is concerned. To get more accurate results, it would be necessary to design/employ proper correlation functionals suited for confined systems.
It is seen that S r amplifies and S p declines with rise in r c , in both ground and excited states under consideration. Besides, for a particular confinement strength, as Z grows, the state of a system becomes more localized with consequent drop and rise in S r , S p respectively.
For the two family of states arising out of configurations (a) 1sns 3 S (n = 2-4) and (b) 1s3s 3 S, 1s3p 3 P, 1s3d 3 D, in the intermediate and lower r c region, the information entropies show interesting crossovers, and finally reach their free-atom limit at certain large r c . In all cases, S t bound is maintained. The emergence of these novel characteristics of S r , S p and S t makes such information-centric analyses valuable tools for structure and dynamics under constrained environment. It would be worthwhile to extend the present study to the case of supposedly more realistic penetrable boundary. Besides we are also interested in several other information measures like Fisher information, Onicescu energy, Complexity etc., in such systems. Some of these works may be undertaken in future.