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

Sangita Majumdar  Amlan K. Roy  Corresponding author. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS> Department of Chemical Sciences
Indian Institute of Science Education and Research (IISER) Kolkata,
Mohanpur-741246, Nadia, WB, India

###### Abstract

Shannon entropy in position ($S_{\mathrm{\mathbf{r}}}$) and momentum
($S_{\mathrm{\mathbf{p}}}$) spaces, along with their sum ($S_{t}$) are
presented for unit-normalized densities of He, Li+ and Be2+ 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, _viz.,_ 1sns (n $=$ 2-4) 3S, 1snp (n $=$ 2-3) 3P, 1s3d 3D 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, _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.

Keywords: Shannon Entropy, quantum confinement, impenetrable boundary, excited
states, Helium-like ions, exchange-correlation.

##  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 $\Omega$ 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 jaskolski96 ; dolmatov04 ; sabin09 ;
sen14 ; leykoo18  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 michels37  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 ludena77 ; marin91 ;
goldman92 ; aquino95 ; sen02 ; laughlin02 ; laughlin04 ; burrows05 ; sen06 ;
burrows06 ; aquino07 ; baye08 ; ciftci09 ; montgomery09 ; roy15 ; roy16 . In
recent years, exact solution of the Schrödinger equation has been found in
terms of Kummer-M function (confluent hypergeometric) burrows06 . 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 reduced 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.

An analogous study of compressed He atom is a prototypical non-trivial
mathematical problem. Due to the presence of inter-electronic repulsion, the
$SU(3)$ symmetry of simplified one-electron case is broken which promises many
exciting physics. Ever since the variational calculation of energy variation
tenseldam52  with respect to cage radius and function of pressure, vigorous
attempts have been known. Some of them include Roothaan-HF-type calculation
with Slater-type basis ludena78  or with its modifications garza12 ,
configuration interaction ludena79 ; rivelino01 , quantum Monte Carlo joslin92
, a host of direct variational schemes with appropriate choice of cut-off
function marin91 ; banerjee06 ; flores-riveros10 ; lesech11 , variational
method with B-splines basis ting-yun01 , Rayleigh-Schrödinger perturbation
theory flores-riveros10 ; montgomery10 . Some other prominent works are
explicitly correlated Hylleraas-type wave functions within variational
framework aquino03 ; flores-riveros08 ; laughlin09 ; wilson10 ; montgomery13 ;
bhattacharyya13 ; montgomery15 ; saha16 , a combination of quantum genetic
algorithm and HF method yakar11 , variational Monte Carlo doma12 ; sarsa11 ,
HF calculation employing local and global basis sets young16  and so on.
Whereas a vast majority of publications have focused on lowest state, low-
lying excited states were also treated quite decently. For example, 1sns 1,3S
states in banerjee06 ; flores-riveros08 ; flores-riveros10 ; yakar11 ; sarsa11
; montgomery13 ; bhattacharyya13 ; montgomery15 ; saha16 ; pupyshev17 , 1s2p
3P, 1P states in banerjee06 ; yakar11 ; pupyshev17 , singly excited 1s3d 3D,
1D and some doubly excited states in yakar11 ; pupyshev17  etc., using a host
of theoretical approaches giving results of varied accuracy.

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) parr89 ;
fiolhais03 ; engel11 . Thus within an exchange-only framework (using two
exchange functionals, _viz._ , local density approximation (LDA) parr89 , and
Becke-88 exchange potential becke88a ), the desired Kohn-Sham (KS) equation
was solved satisfying the Dirichlet boundary condition for many-electron
systems via numerical shooting method garza98 . 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 francisco19 . In another attempt, ground and 1s2s 3S, 1S
states of confined He atom were reported aquino06  taking into account the
LDA-approximated exchange-correlation (XC) (with Perdew-Wang parametrization
for correlation perdew92 ) 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
waugh10 . In a recent work vyboishchikov15 , 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 mayra . 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 vyboishchikov16 ;
vyboishchikov17 . The calculation of static polarizability of confined He and
Ne atoms was done through time-dependent DFT in faassen09 .

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 localization-
delocalization, 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
($S$) shannon51 ; bbi75 , 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 ou19ijqc ), but in _confined_ quantum systems as treated here,
analogous studies are quite limited. Two such reports sen05 ; jiao17  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 mukherjee18 ;
mukherjee18a  for $\ell=0$ as well as non-zero-$\ell$ states. One finds that,
confinement affects $S$ more profoundly in the stronger regime. Further,
$S_{\mathrm{\mathbf{r}}}$ increases with rise in $r_{c}$ and at very
low-$r_{c}$ region ($\approx 0.1$), CHA displays exactly opposite trend from a
free H atom ($S_{\mathrm{\mathbf{r}}}$ 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+,
Be2+), $S$ has been reported for only ground states sen05  by means of BLYP
calculations.

Some reports aquino13 ; cruz16 ; ou17 ; ou17cpl ; rodriguez18 ; francisco19 ;
martinez19  are available on $S$ in _penetrable_ confinement in atoms. For
example, it was observed aquino13  that, in CHA, up to a certain value of
$r_{c}$, $S_{\mathrm{\mathbf{r}}}$ decreases with $r_{c}$. However, for small
$r_{c}$ and depending on barrier height, $S_{\mathrm{\mathbf{r}}}$ may also
increase. Apart from constant potential, it was probed martinez19  for
confinements imposed by a dielectric continuum and by isotropic harmonic
potential. It was also proposed rodriguez18  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 ou17 . Confinement by an inert geometric planar boundary with
finite barrier height has been studied cruz16  within a Thomas-Fermi-Dirac-
Weizsäcker-type DFT framework. Some limited works exist on excited states as
well, _viz._ , low-lying singly ou17  and doubly ou17cpl  excited states. It
is worth noting that, $S$ values in He are available restrepo  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 He-like 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
brual78  and somewhat involved nonlinear Lee-Yang-Parr (LYP) lee88
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 $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}}$
and total Shannon entropy sum
($S=S_{\mathrm{\mathbf{r}}}+S_{\mathrm{\mathbf{p}}}$) with respect to $r_{c}$
is offered for He, Li+, Be2+. Apart from ground state, we also consider singly
excited 3S, 3P, 3D states arising out of configurations 1sns (n $=$ 2-4) 3S,
1snp (n $=$ 2-3) 3P 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 roy97 ; roy97a ; roy97b ; roy02 ; roy04 ; roy05 ; roy07 .
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),

$\mathrm{\mathbf{H}}(\mathrm{\mathbf{r}})\phi_{i}(\mathrm{\mathbf{r}})=\epsilon_{i}(\mathrm{\mathbf{r}})\phi_{i}(\mathrm{\mathbf{r}}),$
(1)

where $\mathrm{\mathbf{H}}$ is the perturbed KS Hamiltonian, written as,

$\displaystyle\mathrm{\mathbf{H}}(\mathrm{\mathbf{r}})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\nabla^{2}+v_{eff}(\mathrm{\mathbf{r}})$
$\displaystyle v_{eff}(\mathrm{\mathbf{r}})$ $\displaystyle=$ $\displaystyle
v_{ne}(\mathrm{\mathbf{r}})+\int\frac{\rho(\mathrm{\mathbf{r}}^{\prime})}{|\mathrm{\mathbf{r}}-\mathrm{\mathbf{r}}^{\prime}|}\mathrm{d}\mathrm{\mathbf{r}}^{\prime}+\frac{\delta
E_{xc}[\rho(\mathrm{\mathbf{r}})]}{\delta\rho(\mathrm{\mathbf{r}})}+v_{conf}(\mathrm{\mathbf{r}}).$
(2)

In the above, $v_{ne}(\mathrm{\mathbf{r}})$ and
$v_{conf}(\mathrm{\mathbf{r}})$ 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),

$v_{conf}(\mathrm{\mathbf{r}})=\begin{cases}0,\ \ \ \ \ \ \ r\leq r_{c}\\\
+\infty,\ \ \ \ r>r_{c}.\\\ \end{cases}$ (3)

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 sahni90
; sahni92 . Accordingly, exchange energy is interpreted as the interaction
energy between an electron at $\mathrm{\mathbf{r}}$ and its Fermi-Coulomb hole
charge density $\rho_{x}(\mathrm{\mathbf{r}},\mathrm{\mathbf{r}}^{\prime})$ at
$\mathrm{\mathbf{r}}^{\prime}$, and given by,

$E_{x}[\rho(\mathrm{\mathbf{r}})]=\frac{1}{2}\int\int\frac{\rho(\mathrm{\mathbf{r}})\rho_{x}(\mathrm{\mathbf{r}},\mathrm{\mathbf{r}}^{\prime})}{|\mathrm{\mathbf{r}}-\mathrm{\mathbf{r}}^{\prime}|}\
\mathrm{d}\mathrm{\mathbf{r}}\ \mathrm{d}\mathrm{\mathbf{r}}^{\prime}.$ (4)

Assuming that a unique local exchange potential $v_{x}(\mathrm{\mathbf{r}})$
exists for a given state, it can be defined as the work done in bringing an
electron to the point $\mathrm{\mathbf{r}}$ against the electric field
generated by its Fermi-Coulomb hole density, leading to the following form,

$v_{x}(\mathrm{\mathbf{r}})=-\int_{\infty}^{r}\mathcal{E}_{x}(\mathrm{\mathbf{r}})\mathrm{d}l,$
(5)

where the electric field is expressed as,

$\mathcal{E}_{x}(\mathrm{\mathbf{r}})=\int\frac{\rho_{x}(\mathrm{\mathbf{r}},\mathrm{\mathbf{r}}^{\prime})(\mathrm{\mathbf{r}}-\mathrm{\mathbf{r}}^{\prime})}{|\mathrm{\mathbf{r}}-\mathrm{\mathbf{r}}^{\prime}|^{3}}\
\mathrm{d}{\mathrm{\mathbf{r}}^{{}^{\prime}}}.$ (6)

The Fermi hole can be written in terms of orbitals as,

$\rho_{x}(\mathrm{\mathbf{r}},\mathrm{\mathbf{r}}^{\prime})=-\frac{\left|\gamma(\mathrm{\mathbf{r}},\mathrm{\mathbf{r}}^{\prime})\right|^{2}}{2\rho(\mathrm{\mathbf{r}})},$
(7)

where
$\left|\gamma(\mathrm{\mathbf{r}},\mathrm{\mathbf{r}}^{\prime})\right|=\sum_{i}\phi_{i}^{*}(\mathrm{\mathbf{r}})\phi_{i}(\mathrm{\mathbf{r}}^{\prime})$
is the single-particle density matrix and $\rho(\mathrm{\mathbf{r}})$ is the
electron density, expressed in terms of occupied atomic orbitals ($n_{i}$
denotes occupation number) as,

$\rho(\mathrm{\mathbf{r}})=\sum_{i=1}^{N}n_{i}|\phi_{i}(\mathrm{\mathbf{r}})|^{2}.$
(8)

While the exchange potential $v_{x}(\mathrm{\mathbf{r}})$ corresponding to a
given state arising from an electronic configuration can be accurately
calculated by the above procedure as delineated, the correlation potential
$v_{c}(\mathrm{\mathbf{r}})$ is unknown and must be approximated for practical
calculations. The current work incorporates two correlation functionals,
namely, a Wigner-type brual78  and LYP lee88 . These two functionals have been
chosen on the basis of their success in the _unconfined_ atomic excited
states, which are recorded in the references roy97 ; roy97a ; roy97b ; roy02 ;
roy04 ; roy05 ; roy07 . 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 $v_{x}(\mathrm{\mathbf{r}})$ and
$v_{c}(\mathrm{\mathbf{r}})$, the resulting KS differential equation,

$\left[-\frac{1}{2}\nabla^{2}+v_{eff}(\mathrm{\mathbf{r}})\right]\phi_{i}(\mathrm{\mathbf{r}})=\varepsilon_{i}\phi_{i}(\mathrm{\mathbf{r}}),$
(9)

needs to be solved, where $v_{eff}(\mathrm{\mathbf{r}})$ 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 roy02 ; roy04 ; roy05 ; roy07 ; roy04a ; roy04b ; roy05a ; roy05b
such as, Coulomb, Húlthen, Yukawa, logarithmic, spiked oscillator, Hellmann
potential, etc., along with its recent extension to quantum confinement sen06
; roy15 ; roy16 . 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 in the interval $[-1,1]$ by an $N$th-order polynomial
$f_{N}(x)$,

$f(x)\cong f_{N}(x)=\sum_{j=0}^{N}f(x_{j})g_{j}(x),$ (10)

which ensures that the approximation be _exact_ at the _collocation points_
$x_{j}$,

$f_{N}(x_{j})=f(x_{j}).$ (11)

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,

$P_{N}^{{}^{\prime}}(x_{j})=0.$ (12)

In Eq. (10), $g_{j}(x)$ are termed _cardinal functions_ , and as such,
expressed as,

$g_{j}(x)=-\frac{1}{N(N+1)P_{N}(x_{j})}\
\frac{(1-x^{2})P_{N}^{{}^{\prime}}(x)}{(x-x_{j})},$ (13)

fulfilling the unique property that,
$g_{j}(x_{j^{{}^{\prime}}})=\delta_{j^{{}^{\prime}},j}$. 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,

$\displaystyle\xi(\mathrm{\mathbf{p}})$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi}\right)^{3/2}\int\phi(\mathrm{\mathbf{r}})\
e^{i{\mathrm{\mathbf{p}}}.{\mathrm{\mathbf{r}}}}\mathrm{d}{\mathrm{\mathbf{r}}}.$
(14)

It is to be noted here that $\xi(\mathrm{\mathbf{p}})$ is not normalized;
hence needs to be normalized. The normalized $r$\- and $p$-space densities are
represented as
$\rho(\mathrm{\mathbf{r}})=\sum_{i=1}^{N}n_{i}|\phi_{i}(\mathrm{\mathbf{r}})|^{2}$
and
$\Pi(\mathrm{\mathbf{p}})=\sum_{i=1}^{N}n_{i}|\xi_{i}(\mathrm{\mathbf{p}})|^{2}$
respectively, where $n_{i}$ represents the occupation number of each orbital.

Next $S_{\mathrm{\mathbf{r}}},~{}S_{\mathrm{\mathbf{p}}}$ 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,

$\displaystyle S_{\mathrm{\mathbf{r}}}$
$\displaystyle=-\int_{{\mathcal{R}}^{3}}\rho(\mathrm{\mathbf{r}})\
\ln[\rho(\mathrm{\mathbf{r}})]\ \mathrm{d}\mathrm{\mathbf{r}},\ \ \
S_{\mathrm{\mathbf{p}}}=-\int_{{\mathcal{R}}^{3}}\Pi(\mathrm{\mathbf{p}})\
\ln[\Pi(\mathrm{\mathbf{p}})]\ \mathrm{d}\mathrm{\mathbf{p}},$ (15)
$\displaystyle S_{t}$
$\displaystyle=\left[S_{\mathrm{\mathbf{r}}}+S_{\mathrm{\mathbf{p}}}\right]\geq
3(1+\ln\pi),\ \ \ \ \ \text{in 3 dimension}.$

Here $\rho(\mathrm{\mathbf{r}})$ and $\Pi(\mathrm{\mathbf{p}})$ 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 $r_{c}\rightarrow\infty$ limit. All results given in
the following tables and plots have been checked for above convergence.

Table 1:  $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ (a.u.) in
ground states of confined He, Li+, Be2+. See text for details.

Species | $r_{c}$ | X-only | XC-Wigner | XC-LYP | Literature11footnotemark: 1 |  |
---|---|---|---|---|---|---|---
|  | Sr | Sp | St | Sr | Sp | St | Sr | Sp | St | Sr | Sp | St
| 0.1 |  $-$6.2534 | 12.855 | 6.5744 |  $-$6.2534 | 12.855 | 6.5744 |  $-$6.2534 | 12.855 | 6.5744 |  |  |
| 0.3 |  $-$3.004 | 9.580 | 6.576 |  $-$3.004 | 9.580 | 6.576 |  $-$3.004 | 9.580 | 6.576 |  $-$2.988 | 9.488 | 6.5
| 0.5 |  $-$1.525 | 8.075 | 6.550 |  $-$1.525 | 8.075 | 6.550 |  $-$1.525 | 8.075 | 6.550 |  $-$1.498 | 8.023 | 6.525
| 1 | 0.389 | 6.118 | 6.507 | 0.387 | 6.119 | 6.506 | 0.388 | 6.118 | 6.506 | 0.4326 | 6.078 | 6.51
He22footnotemark: 2,33footnotemark: 3 | 1.4 | 1.22 | 5.273 | 6.493 | 1.22 | 5.276 | 6.49 | 1.22 | 5.327 | 6.547 | 1.2739 | 5.234 | 6.508
| 2 | 1.97 | 4.547 | 6.517 | 1.96 | 4.555 | 6.515 | 1.96 | 4.549 | 6.509 | 2.0097 | 4.519 | 6.528
| 3 | 2.50 | 4.061 | 6.561 | 2.48 | 4.080 | 6.560 | 2.50 | 4.068 | 6.568 | 2.5241 | 4.057 | 6.581
| 4 | 2.65 | 3.943 | 6.593 | 2.63 | 3.969 | 6.599 | 2.64 | 3.952 | 6.592 | 2.6651 | 3.945 | 6.61
| 5 | 2.68 | 3.921 | 6.608 | 2.65 | 3.95 | 6.60 | 2.67 | 3.932 | 6.602 | 2.7042 | 3.918 | 6.622
| 6 | 2.69 | 3.918 | 6.608 | 2.66 | 3.95 | 6.61 | 2.68 | 3.93 | 6.61 | 2.7106 | 3.914 | 6.625
| 7 | 2.69 | 3.918 | 6.608 | 2.66 | 3.95 | 6.61 | 2.68 | 3.93 | 6.61 | 2.7117 | 3.913 | 6.625
| 0.1 |  $-$6.2665 | 12.860 | 6.5935 |  $-$6.2665 | 12.860 | 6.5935 |  $-$6.2665 | 12.860 | 6.5935 |  |  |
| 0.3 |  $-$3.050 | 9.604 | 6.554 |  $-$3.050 | 9.604 | 6.554 |  $-$3.050 | 9.604 | 6.554 |  $-$3.034 | 9.538 | 6.504
| 0.8 |  $-$0.392 | 6.890 | 6.498 |  $-$0.393 | 6.891 | 6.498 |  $-$0.392 | 6.890 | 6.498 |  $-$0.353 | 6.849 | 6.496
| 1 | 0.12 | 6.376 | 6.496 | 0.121 | 6.378 | 6.499 | 0.122 | 6.377 | 6.499 | 0.1659 | 6.335 | 6.501
Li+44footnotemark: 4,55footnotemark: 5 | 2 | 1.135 | 5.431 | 6.566 | 1.12 | 5.440 | 6.56 | 1.13 | 5.43 | 6.56 | 1.174 | 5.4 | 6.574
| 2.5 | 1.22 | 5.361 | 6.581 | 1.21 | 5.373 | 6.58 | 1.22 | 5.36 | 6.58 | 1.2618 | 5.331 | 6.593
| 3 | 1.24 | 5.346 | 6.586 | 1.23 | 5.358 | 6.58 | 1.24 | 5.34 | 6.58 | 1.2878 | 5.313 | 6.601
| 4 | 1.25 | 5.343 | 6.593 | 1.23 | 5.355 | 6.58 | 1.24 | 5.34 | 6.58 | 1.2942 | 5.309 | 6.603
| 7 | 1.25 | 5.343 | 6.593 | 1.23 | 5.355 | 6.58 | 1.24 | 5.34 | 6.58 |  |  |
| 0.1 |  $-$6.2801 | 12.866 | 6.5859 |  $-$6.2801 | 12.866 | 6.5859 |  $-$6.2801 | 12.866 | 6.5859 |  |  |
| 0.3 |  $-$3.102 | 9.636 | 6.534 |  $-$3.102 | 9.636 | 6.534 |  $-$3.102 | 9.636 | 6.534 |  |  |
| 0.5 |  $-$1.725 | 8.226 | 6.501 |  $-$1.725 | 8.226 | 6.501 |  $-$1.725 | 8.226 | 6.501 |  |  |
| 1 |  $-$0.229 | 6.748 | 6.519 |  $-$0.231 | 6.750 | 6.519 |  $-$0.229 | 6.748 | 6.519 |  |  |
Be2+ | 1.5 | 0.191 | 6.373 | 6.564 | 0.186 | 6.378 | 6.564 | 0.190 | 6.374 | 6.564 |  |  |
| 2 | 0.27 | 6.311 | 6.581 | 0.26 | 6.317 | 6.577 | 0.26 | 6.312 | 6.572 |  |  |
| 2.5 | 0.28 | 6.305 | 6.585 | 0.27 | 6.311 | 6.581 | 0.26 | 6.306 | 6.566 |  |  |
| 3 | 0.28 | 6.305 | 6.585 | 0.27 | 6.311 | 6.581 | 0.26 | 6.306 | 6.566 |  |  |
| 7 | 0.28 | 6.305 | 6.585 | 0.27 | 6.311 | 6.581 | 0.26 | 6.306 | 6.566 |  |  |

aDFT calculation sen05 , using BLYP functional.
---
bFor free atom, using $N$-normalized HF density–$S_{\mathrm{\mathbf{r}}}$:
4.01 rodriguez18 , 4.06 gadre85 , 4.0100 amovilli , 4.0107 francisco19 ;
$S_{\mathrm{\mathbf{p}}}$: 6.45 gadre85 ; $S_{t}$: 10.52 gadre85 .
Our X-only results are 4.00, 6.45 and 10.52 respectively.
cCorrelated $S_{\mathrm{\mathbf{r}}}$ values in free He atom: (i) unit-normalized density: 2.7051028 lin15 , 2.705 ou19  | (ii) N-normalized density: 4.40106,
(Variational Monte Carlo), 4.0256 (Diffusion Monte Carlo) amovilli . Our XC
results are 3.93 (XC-Wigner) and 3.96 (XC-LYP).
dHF $S_{\mathrm{\mathbf{r}}}$ value in free Li+ ion using N-normalized
density: 1.1023 amovilli . Our X-only result is 1.11.
eCorrelated $S_{\mathrm{\mathbf{r}}}$ value in Li+ ion: (i) unit-normalized
density: 1.2552726 lin15  (ii) N-normalized density: 1.1204 (Variational
Monte Carlo), amovilli . 1.1143 (Diffusion Monte Carlo). Our XC results are:
1.08 (XC-Wigner), 1.10 (XC-LYP).

##  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 (i) radial and (ii) angular contributions. The
angular part remains invariant in both spaces with respect to change in
boundary condition resulting from confinement. Note that, an analogous energy
analysis with respect to $r_{c}$ in these and other excited states, would be
presented elsewhere. However, it suffices here to mention that, energies for
both ground and excited states obtained with the present method are in good
agreement with those available in literature. Here our aim is to investigate
$S$ for confined two-electron atomic systems at various $r_{c}$ in several
low-lying states. These are calculated for spherically confined He and
extended to confined iso-electronic members, namely, Li+ and Be2+. Apart from
ground state, following low-lying singly excited states have been considered,
_viz._ , 1sns 3S with n$=$ 2-4; 1snp 3P with n $=$ 2-3; 1snd 3D having n $=$
3\. The necessary results are presented in the following tables and plots
along with available literature values, for a comparative discussion. All
calculations are performed with unit-normalized density.

Figure 1: Variation of $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$
for He-isoelectronic series $(Z=2-4)$ with $r_{c}$ in panels (a)–(c).

To start with, Table I imprints the numerical results of
$S_{\mathrm{\mathbf{r}}}$, $S_{\mathrm{\mathbf{p}}},S_{t}$ for confined He,
Li+ and Be2+ in their ground states. In order to put things in perspective,
here and in all other tables, three sets of calculations have been performed
at each $r_{c}$, namely, (i) exchange-only (ii) XC with Wigner correlation
(iii) XC with LYP correlation. Throughout the article, these three results are
denoted by labels X-only, XC-Wigner and XC-LYP respectively. This can give us
an idea how X-only and HF results compare and contrast. Moreover, it will help
us in getting a sense of correlation contribution in current context,
approximated by two functionals. In all three occasions (i)–(iii),
$S_{\mathrm{\mathbf{r}}}$’s increase with rise in $r_{c}$ and finally merge to
corresponding free atom entropy at a sufficiently large $r_{c}$. The reported
values of $S_{\mathrm{\mathbf{r}}}$ in this table further reinforces the
previous conclusions sen05 ; rodriguez18  that impenetrable walls impose
confinements in a way that localizes the electron density, and consequently
$S_{\mathrm{\mathbf{r}}}\\!\rightarrow\\!-\infty$ when $r_{c}\rightarrow 0$.
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, $S_{\mathrm{\mathbf{r}}}$ decreases. On the
contrary, $S_{\mathrm{\mathbf{p}}}$ 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 bbi75 . The qualitative behavior of
$S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ with growth in $r_{c}$
does not change much with atomic charge ($Z$), although their numerical values
differ. In fact, at a given $r_{c}$, $S_{\mathrm{\mathbf{r}}}$ regresses and
$S_{\mathrm{\mathbf{p}}}$ 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 guevara03 . 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 ($\leq
0.5$), _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 waugh10 . For He and Li+, these
have been estimated by BLYP calculation sen05  in most of the $r_{c}$’s
considered here, which are appropriately quoted. Note that the HF values
gadre85  for $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ 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
$S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ values of He and Li+
within HF method gadre85 ; amovilli , 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 $S_{\mathrm{\mathbf{r}}}$ have been presented rodriguez18 . The
same has also been calculated from a DFT-based study with hybrid exchange
functional francisco19 . 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 $S_{\mathrm{\mathbf{r}}}$
in $U_{0}\rightarrow\infty$ corresponds to the impenetrable confinement.
Highly accurate benchmark-quality result for $S_{\mathrm{\mathbf{r}}}$ was
calculated from a Hylleraas-type variational method producing a value of
2.7051028 and 1.2552726 for free He and Li+ lin15  respectively. Correlated
results obtained from variational Monte carlo and diffusion Monte Carlo
methods amovilli  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
Be2+.

Table 2:  $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ (a.u.) in
1s2s 3S states of confined He, Li+, Be2+. See text for details.

Species | $r_{c}$ | X-only | XC-Wigner | XC-LYP
---|---|---|---|---
|  | Sr | Sp | St | Sr | Sp | St | Sr | Sp | St
| 0.1 |  $-$6.2172 | 14.190 | 7.9728 |  $-$6.2172 | 14.190 | 7.9728 |  $-$6.2172 | 14.190 | 7.9728
| 0.5 |  $-$1.4472 | 9.376 | 7.9288 |  $-$1.4472 | 9.376 | 7.9288 |  $-$1.4472 | 9.3756 | 7.9288
| 1 | 0.547 | 7.333 | 7.880 | 0.547 | 7.333 | 7.880 | 0.547 | 7.333 | 7.880
| 2 | 2.417 | 5.409 | 7.826 | 2.415 | 5.410 | 7.825 | 2.417 | 5.409 | 7.826
He11footnotemark: 1 | 4 | 3.956 | 3.86 | 7.819 | 3.948 | 3.87 | 7.818 | 3.953 | 3.86 | 7.813
| 6 | 4.60 | 3.22 | 7.82 | 4.59 | 3.23 | 7.82 | 4.59 | 3.23 | 7.82
| 6.5 | 4.71 | 3.11 | 7.82 | 4.69 | 3.13 | 7.82 | 4.69 | 3.13 | 7.82
| 7.5 | 4.87 | 2.94 | 7.81 | 4.85 | 2.96 | 7.81 | 4.85 | 2.97 | 7.82
| 8.5 | 4.99 | 2.81 | 7.80 | 4.96 | 2.84 | 7.80 | 4.96 | 2.85 | 7.81
| 10 | 5.10 | 2.69 | 7.79 | 5.06 | 2.72 | 7.78 | 5.03 | 2.75 | 7.78
| 0.1 |  $-$6.2246 | 14.191 | 7.9664 |  $-$6.2246 | 14.191 | 7.9664 |  $-$6.2246 | 14.191 | 7.9664
| 0.5 |  $-$1.4905 | 9.392 | 7.9015 |  $-$1.4905 | 9.392 | 7.9015 |  $-$1.4905 | 9.392 | 7.9015
| 1 | 0.442 | 7.401 | 7.843 | 0.442 | 7.401 | 7.843 | 0.442 | 7.401 | 7.843
| 1.5 | 1.481 | 6.335 | 7.816 | 1.480 | 6.336 | 7.816 | 1.481 | 6.336 | 7.817
Li+ | 2 | 2.141 | 5.669 | 7.810 | 2.138 | 5.671 | 7.809 | 2.140 | 5.669 | 7.809
| 3 | 2.910 | 4.90 | 7.810 | 2.905 | 4.90 | 7.805 | 2.909 | 4.905 | 7.814
| 4 | 3.32 | 4.49 | 7.81 | 3.312 | 4.49 | 7.802 | 3.31 | 4.49 | 7.80
| 7 | 3.70 | 4.07 | 7.77 | 3.68 | 4.09 | 7.77 | 3.69 | 4.09 | 7.78
| 8.5 | 3.72 | 4.05 | 7.77 | 3.70 | 4.07 | 7.77 | 3.70 | 4.07 | 7.77
| 10 | 3.72 | 4.05 | 7.77 | 3.70 | 4.07 | 7.77 | 3.70 | 4.07 | 7.77
| 0.1 |  $-$6.2320 | 14.191 | 7.9590 |  $-$6.2320 | 14.191 | 7.9590 |  $-$6.2320 | 14.191 | 7.9590
| 0.5 |  $-$1.5376 | 9.415 | 7.8774 |  $-$1.537 | 9.415 | 7.8774 |  $-$1.5376 | 9.415 | 7.8774
| 1 | 0.322 | 7.499 | 7.821 | 0.322 | 7.500 | 7.822 | 0.322 | 7.499 | 7.821
| 2 | 1.829 | 5.981 | 7.810 | 1.826 | 5.983 | 7.809 | 1.828 | 5.981 | 7.809
Be2+ | 3 | 2.42 | 5.38 | 7.80 | 2.141 | 5.39 | 7.531 | 2.418 | 5.38 | 7.798
| 4 | 2.66 | 5.12 | 7.78 | 2.651 | 5.13 | 7.781 | 2.656 | 5.12 | 7.776
| 5 | 2.73 | 5.03 | 7.76 | 2.72 | 5.05 | 7.77 | 2.72 | 5.04 | 7.76
| 6 | 2.75 | 5.02 | 7.77 | 2.73 | 5.03 | 7.76 | 2.74 | 5.02 | 7.76
| 20 | 2.75 | 5.02 | 7.77 | 2.73 | 5.03 | 7.76 | 2.74 | 5.02 | 7.76

aCorrelated $S_{\mathrm{\mathbf{r}}}$ value in free atom is: 5.239 ou19 .
---

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 $S_{\mathrm{\mathbf{r}}}$,
$S_{\mathrm{\mathbf{p}}}$ 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 $Z$,
$S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}}$ 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 $S_{\mathrm{\mathbf{r}}}$ and $S_{\mathrm{\mathbf{p}}}$
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 $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}}$ 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 sen05 .

Figure 2: Variation of $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}}$ in
1sns 3S ($n=2-4$) states of confined He with $r_{c}$. See text for details.

Now we move on to some low-lying singly excited
state-$S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ for He, Li+ and
Be2+ in Tables II-IV. Following the presentation strategy of previous table,
it reports results for 1s2s 3S, 1s2p 3P and 1s3d 3D states respectively. Like
the ground state, here also in all three states, both X-only and correlation-
included $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}}$ moves up and down
respectively with growth in $r_{c}$. In all cases again $S_{t}$ obeys the
stipulated lower bound bbi75 . The qualitative pattern of
$S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ 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 $r_{c}$,
$S_{\mathrm{\mathbf{r}}}$ falls off and $S_{\mathrm{\mathbf{p}}}$ 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 $S_{\mathrm{\mathbf{r}}}$ in 1s2s 3S 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.

Table 3:  $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ (a.u.) in
1s2p 3P states of confined He, Li+, Be2+. See text for details.

Species | $r_{c}$ | X-only | XC-Wigner | XC-LYP
---|---|---|---|---
|  | Sr | Sp | St | Sr | Sp | St | Sr | Sp | St
| 0.5 |  $-$1.392 | 8.596 | 7.204 |  $-$1.392 | 8.596 | 7.204 |  $-$1.392 | 8.596 | 7.204
| 0.8 |  $-$0.031 | 7.213 | 7.182 |  $-$0.031 | 7.213 | 7.182 |  $-$0.031 | 7.213 | 7.182
| 1 | 0.60 | 6.570 | 7.170 | 0.60 | 6.570 | 7.170 | 0.60 | 6.570 | 7.170
| 3 | 3.26 | 4.04 | 7.30 | 3.26 | 4.05 | 7.31 | 3.26 | 4.04 | 7.30
He11footnotemark: 1 | 5 | 4.09 | 3.49 | 7.58 | 4.08 | 3.50 | 7.58 | 4.08 | 3.50 | 7.58
| 6 | 4.35 | 3.34 | 7.69 | 4.32 | 3.36 | 7.68 | 4.33 | 3.35 | 7.68
| 7 | 4.54 | 3.23 | 7.77 | 4.51 | 3.25 | 7.76 | 4.51 | 3.25 | 7.76
| 7.6 | 4.63 | 3.18 | 7.81 | 4.58 | 3.21 | 7.79 | 4.58 | 3.21 | 7.79
| 8 | 4.68 | 3.15 | 7.83 | 4.65 | 3.18 | 7.83 | 4.64 | 3.18 | 7.82
| 10 | 4.86 | 3.04 | 7.90 | 4.82 | 3.08 | 7.90 | 4.78 | 3.10 | 7.88
| 0.5 |  $-$1.437 | 8.620 | 7.183 |  $-$1.437 | 8.620 | 7.183 |  $-$1.437 | 8.620 | 7.183
| 0.8 |  $-$0.121 | 7.284 | 7.163 |  $-$0.121 | 7.284 | 7.163 |  $-$0.121 | 7.284 | 7.163
| 1 | 0.471 | 6.693 | 7.164 | 0.470 | 6.694 | 7.164 | 0.47 | 6.69 | 7.16
| 1.5 | 1.44 | 5.91 | 7.35 | 1.438 | 5.91 | 7.358 | 1.44 | 5.91 | 7.35
Li+ | 2 | 2.009 | 5.29 | 7.299 | 2.005 | 5.30 | 7.305 | 2.008 | 5.30 | 7.308
| 3 | 2.65 | 4.85 | 7.50 | 2.64 | 4.86 | 7.50 | 2.653 | 4.86 | 7.513
| 6 | 3.33 | 4.47 | 7.80 | 3.31 | 4.49 | 7.80 | 3.31 | 4.48 | 7.79
| 8 | 3.39 | 4.45 | 7.84 | 3.36 | 4.47 | 7.83 | 3.37 | 4.47 | 7.84
| 9 | 3.40 | 4.45 | 7.85 | 3.36 | 4.47 | 7.83 | 3.37 | 4.47 | 7.84
| 10 | 3.40 | 4.45 | 7.85 | 3.37 | 4.47 | 7.84 | 3.37 | 4.47 | 7.84
| 0.5 |  $-$1.490 | 8.658 | 7.168 |  $-$1.490 | 8.658 | 7.168 |  $-$1.490 | 8.658 | 7.168
| 0.8 |  $-$0.235 | 7.402 | 7.167 |  $-$0.235 | 7.402 | 7.167 |  $-$0.235 | 7.402 | 7.167
| 1.2 | 0.698 | 6.532 | 7.23 | 0.697 | 6.534 | 7.231 | 0.698 | 6.532 | 7.23
| 2 | 1.589 | 5.857 | 7.446 | 1.585 | 5.861 | 7.446 | 1.588 | 5.858 | 7.446
Be2+ | 2.5 | 1.892 | 5.670 | 7.562 | 1.88 | 5.67 | 7.55 | 1.890 | 5.701 | 7.591
| 5 | 2.37 | 5.41 | 7.78 | 2.35 | 5.43 | 7.78 | 2.36 | 5.42 | 7.78
| 6 | 2.38 | 5.41 | 7.79 | 2.36 | 5.42 | 7.78 | 2.37 | 5.42 | 7.79
| 10 | 2.38 | 5.41 | 7.79 | 2.36 | 5.42 | 7.78 | 2.37 | 5.42 | 7.79

aCorrelated $S_{\mathrm{\mathbf{r}}}$ value in free atom is: 5.356. restrepo .
---

Table 4:  $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ (a.u.) in
1s3d 3D states of confined He, Li+, Be2+. See text for details.

Species | $r_{c}$ | X-only | XC-Wigner | XC-LYP
---|---|---|---|---
|  | Sr | Sp | St | Sr | Sp | St | Sr | Sp | St
| 0.5 |  $-$1.3053 | 9.110 | 7.8047 |  $-$1.3053 | 9.110 | 7.8047 |  $-$1.3053 | 9.110 | 7.8047
| 1 | 0.707 | 7.073 | 7.780 | 0.706 | 7.073 | 7.779 | 0.707 | 7.073 | 7.780
| 1.5 | 1.828 | 5.938 | 7.766 | 1.826 | 5.939 | 7.765 | 1.827 | 5.938 | 7.765
| 2.6 | 3.155 | 4.640 | 7.795 | 3.150 | 4.645 | 7.795 | 3.153 | 4.641 | 7.794
He11footnotemark: 1 | 4 | 3.960 | 3.957 | 7.917 | 3.951 | 3.967 | 7.918 | 3.957 | 3.960 | 7.917
| 5 | 4.327 | 3.687 | 8.014 | 4.317 | 3.699 | 8.016 | 4.323 | 3.691 | 8.014
| 6 | 4.61 | 3.48 | 8.09 | 4.60 | 3.50 | 8.10 | 4.60 | 3.49 | 8.09
| 7 | 4.85 | 3.33 | 8.18 | 4.83 | 3.34 | 8.17 | 4.83 | 3.34 | 8.17
| 8 | 5.04 | 3.19 | 8.23 | 5.03 | 3.21 | 8.24 | 5.02 | 3.22 | 8.24
| 10 | 5.36 | 2.97 | 8.33 | 5.35 | 2.99 | 8.34 | 5.27 | 3.04 | 8.24
| 0.5 |  $-$1.339 | 9.129 | 7.790 |  $-$1.339 | 9.129 | 7.790 |  $-$1.339 | 9.129 | 7.790
| 0.8 |  $-$0.003 | 7.774 | 7.771 |  $-$0.003 | 7.774 | 7.771 |  $-$0.003 | 7.774 | 7.771
| 1 | 0.601 | 7.161 | 7.762 | 0.601 | 7.161 | 7.762 | 0.601 | 7.161 | 7.762
| 2.5 | 2.611 | 5.27 | 7.881 | 2.608 | 5.280 | 7.888 | 2.611 | 5.277 | 7.888
Li+ | 3 | 2.918 | 5.042 | 7.960 | 2.914 | 5.047 | 7.961 | 2.917 | 5.043 | 7.960
| 4 | 3.373 | 4.72 | 8.093 | 3.36 | 4.73 | 8.09 | 3.37 | 4.72 | 8.09
| 5 | 3.704 | 4.50 | 8.204 | 3.69 | 4.51 | 8.20 | 3.70 | 4.50 | 8.20
| 6.5 | 4.06 | 4.26 | 8.32 | 4.05 | 4.27 | 8.32 | 4.05 | 4.27 | 8.32
| 7 | 4.16 | 4.20 | 8.36 | 4.15 | 4.21 | 8.36 | 4.14 | 4.21 | 8.35
| 7.5 | 4.24 | 4.14 | 8.38 | 4.23 | 4.15 | 8.38 | 4.22 | 4.16 | 8.38
| 10 | 4.53 | 3.97 | 8.50 | 4.51 | 3.99 | 8.50 | 4.46 | 4.03 | 8.49
| 0.5 |  $-$1.3803 | 9.157 | 7.776 |  $-$1.3803 | 9.157 | 7.776 |  $-$1.3803 | 9.157 | 7.776
| 0.8 |  $-$0.097 | 7.857 | 7.760 |  $-$0.097 | 7.857 | 7.760 |  $-$0.097 | 7.857 | 7.760
| 1 | 0.460 | 7.303 | 7.763 | 0.459 | 7.304 | 7.763 | 0.460 | 7.303 | 7.763
| 2.5 | 2.21 | 5.79 | 8.00 | 2.211 | 5.79 | 8.001 | 2.213 | 5.79 | 8.003
Be2+ | 3 | 2.496 | 5.59 | 8.086 | 2.492 | 5.60 | 8.092 | 2.495 | 5.59 | 8.085
| 4 | 2.911 | 5.32 | 8.231 | 2.907 | 5.32 | 8.227 | 2.909 | 5.32 | 8.229
| 7 | 3.53 | 4.94 | 8.47 | 3.51 | 4.96 | 8.47 | 3.51 | 4.96 | 8.47
| 8 | 3.60 | 4.91 | 8.51 | 3.59 | 4.92 | 8.51 | 3.58 | 4.93 | 8.51
| 8.5 | 3.63 | 4.90 | 8.53 | 3.61 | 4.92 | 8.53 | 3.60 | 4.92 | 8.52
| 10 | 3.66 | 4.90 | 8.62 | 3.64 | 4.91 | 8.55 | 3.62 | 4.92 | 8.54

aCorrelated $S_{\mathrm{\mathbf{r}}}$ value in free atom is: 6.634 restrepo .
---

In order to gain a better understand of the effect of confinement on excited
states, $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}}$ 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, $S_{\mathrm{\mathbf{r}}}$ gains with rise in
$r_{c}$, while $S_{\mathrm{\mathbf{p}}}$ declines. Now for a fixed $r_{c}$,
the behavior of $S_{\mathrm{\mathbf{r}}}$ 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,
$S_{\mathrm{\mathbf{r}}}$ 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 $S_{\mathrm{\mathbf{r}}}$ 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 $S_{\mathrm{\mathbf{r}}}$ for different states occurs. From
the inset plots of panel (a) it is noticed that, at $r_{c}\approx 7.43$ there
is a crossing between $S_{\mathrm{\mathbf{r}}}^{(1s2s)}$ and
$S_{\mathrm{\mathbf{r}}}^{(1s4s)}$; another crossing occurs at $r_{c}\approx
6.36$ between $S_{\mathrm{\mathbf{r}}}^{(1s3s)}$ and
$S_{\mathrm{\mathbf{r}}}^{(1s4s)}$. With further gain in confinement strength,
at $r_{c}\approx 0.33$ and 0.34, crossings take place between
$S_{\mathrm{\mathbf{r}}}^{(1s2s)},S_{\mathrm{\mathbf{r}}}^{(1s4s)}$ and
$S_{\mathrm{\mathbf{r}}}^{(1s2s)},S_{\mathrm{\mathbf{r}}}^{(1s3s)}$
respectively. Hence, one encounters frequent change in the order arrangement
on proceeding from free to strong confined regime. In the strong confinement
regime ($r_{c}\approx 0.1$) the following ordering of entropy holds good:
$S_{\mathrm{\mathbf{r}}}^{(1s2s)}>S_{\mathrm{\mathbf{r}}}^{(1s3s)}>S_{\mathrm{\mathbf{r}}}^{(1s4s)}$.
Apparently, there exists an interplay between two competing effects, namely,
(i) radial confinement (localization) and (ii) accumulation of nodes and humps
with growth in $n$ (delocalization). And these two opposing forces control the
ordering of $S$ values of these states. Similarly, in $p$ space also, such
crossovers prevail at various $r_{c}$’s. However, at $r_{c}\rightarrow\infty$
limit and $r_{c}=0.1$, $S_{\mathrm{\mathbf{p}}}$ shows opposite trend to that
of $S_{\mathrm{\mathbf{r}}}$.

Figure 3: $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}},S_{t}$ in 1s3s 3S,
1s3p 3P, 1s3d 3D states of confined He, against $r_{c}$. See text for details.

As a continuation of the earlier discussion, we present in Fig. 3, X-only
$S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}}$ and $S_{t}$ as a functions
of $r_{c}$ corresponding to 3S, 3P and 3D states resulting from 1s3s, 1s3p and
1s3d configurations of He, in panels (a)–(c). From panel (a), at $r_{c}\approx
10$, we obtain the following ordering of entropy:
$S_{\mathrm{\mathbf{r}}}(^{3}S)>S_{\mathrm{\mathbf{r}}}(^{3}P)>S_{\mathrm{\mathbf{r}}}(^{3}D)$,
indicating a drop in fluctuation as one passes from 3S to 3P to 3D. 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
$S_{\mathrm{\mathbf{r}}}(^{3}P)>S_{\mathrm{\mathbf{r}}}(^{3}S)>S_{\mathrm{\mathbf{r}}}(^{3}D)$
at $(r_{c}\approx 0.1)$, representing strong confinement region. Now in
conjugate space, at higher $r_{c}$ region $\approx 20$,
$S_{\mathrm{\mathbf{p}}}$ displays an exact opposite trend to that of
$S_{\mathrm{\mathbf{r}}}$ 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
$S_{\mathrm{\mathbf{p}}}(^{3}S)>S_{\mathrm{\mathbf{r}}}(^{3}P)>S_{\mathrm{\mathbf{r}}}(^{3}D)$,
at $r_{c}\approx 0.1$. 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 $S_{\mathrm{\mathbf{r}}}$ amplifies and
$S_{\mathrm{\mathbf{p}}}$ 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 $S_{\mathrm{\mathbf{r}}},S_{\mathrm{\mathbf{p}}}$
respectively. For the two family of states arising out of configurations (a)
1sns 3S (n $=$ 2-4) and (b) 1s3s 3S, 1s3p 3P, 1s3d 3D, 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 $S_{\mathrm{\mathbf{r}}}$, $S_{\mathrm{\mathbf{p}}}$ 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.

##  V Acknowledgement

SM is thankful to IISER Kolkata for a Senior Research Fellowship. Financial
support from DAE BRNS, Mumbai (sanction order: 58/14/03/2019-BRNS) is
gratefully acknowledged. Dr. Neetik Mukherjee is thanked for fruitful
discussion. We thank the three anonymous referees for their valuable comments
and suggestions.

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