# Helium-like ions in $d$-dimensions: analyticity and generalized ground state
Majorana solutions

Adrian M. Escobar-Ruiz<EMAIL_ADDRESS>Horacio Olivares-Pilón
<EMAIL_ADDRESS>Norberto Aquino<EMAIL_ADDRESS>Salvador A. Cruz
<EMAIL_ADDRESS>Departamento de Fisica, Universidad Autónoma Metropolitana-
Iztapalapa, San Rafael Atlixco 186, C.P. 09340 Ciudad de México, México

###### Abstract

Non-relativistic Helium-like ions $(-e,-e,Ze)$ with static nucleus in a
$d-$dimensional space $\mathbb{R}^{d}$ ($d>1$) are considered. Assuming
$r^{-1}$ Coulomb interactions, a 2-parametric correlated Hylleraas-type trial
function is used to calculate the ground state energy of the system in the
domain $Z\leq 10$. For odd $d=3,5$, the variational energy is given by a
rational algebraic function of the variational parameters whilst for even
$d=2,4$ it is shown for the first time that it corresponds to a more
complicated non-algebraic expression. This twofold analyticity will hold for
any $d$. It allows us to construct reasonably accurate approximate solutions
for the ground state energy $E_{0}(Z,d)$ in the form of compact analytical
expressions. We call them generalized Majorana solutions. They reproduce the
first leading terms in the celebrated $\frac{1}{Z}$ expansion, and serve as
generating functions for certain correlation-dependent properties. The (first)
critical charge $Z_{\rm c}$ vs $d$ and the Shannon entropy $S_{r}^{(d)}$ vs
$Z$ are also calculated within the present variational approach. In the light
of these results, for the physically important case $d=3$ a more general
3-parametric correlated Hylleraas-type trial is used to compute the finite
mass effects in the Majorana solution for a three-body Coulomb system with
arbitrary charges and masses. It admits a straightforward generalization to
any $d$ as well. Concrete results for the systems $e^{-}\,e^{-}\,e^{+}$,
$H_{2}^{+}$ and $H^{-}$ are indicated explicitly. Our variational analytical
results are in excellent agreement with the exact numerical values reported in
the literature.

##  I Introduction

Ever since the advent of quantum mechanics the description of the helium atom
and isoelectronic ions, which illustrate the properties of few-electron atomic
ions in non-relativistic quantum electrodynamics (QED), has attracted the
interest of physicists and chemists. Solving approximately this well known
system is one of the basic problems in nonrelativistic quantum mechanics: it
appears in many textbooks. Moreover, the helium atom continues to play a key
role as a testing ground for approximate methods, being the electron-electron
repulsion term the cause of non-separability of the Schrödinger equation, and
especially for techniques to assess the uncertainty.

In fact, as early as 1929, Hylleraas Hylleraas  studied the ground state of
the He atom using a very simple correlated trial wavefunction
$\psi=\psi(r_{1},r_{2},r_{12})$ in terms of all three relative mutual
distances among particles. Such an early attempt lead to a more general
approach often referred nowadays under the name of Hylleraas method (see the
review article DrakeG ).

Also, as it is now recognized, during the same time period Ettore Majorana had
advanced work in this direction appearing in his unpublished notes ( see
Esposito  for historical account and further references). In that work
Majorana proposed, for three-dimensional He-like ions, an empirical one-
parameter formula for the ground-state energy versus nuclear charge $Z$ (in
a.u.):

$E\ =\ -Z^{2}\ +\ \frac{5}{8}\,Z\ +\ b\ ,$ (1)

where $b$ is a parameter, which can be found variationally. In fact, the
analytical expression (1) corresponds to the variational ground state energy
with non-correlated Hylleraas function $e^{-Z\,(r_{1}+r_{2})}$. We call this
the 3D Majorana solution.

Since then more accurate treatments of He-like systems in three dimensions
have been reported, such as the use of a modified Hylleraas-like function in
the form of exponentials in all three interparticle distances, employed
originally in variational calculations by Delves and Kalotas Delves  and then
by Thakkar and Smith Thakkar  to compute the ground state energy of He-like
ions. Further efforts to obtain higher accurate calculations have been
developed. Notably, one of the most accurate calculations reported so far
($\sim 35$ significant digits) for the ground-state energy of He-like systems
with static nuclear charges $Z=1–10$ have been reported Nakashima  and efforts
to improve this accuracy still continue Drake-Zong-Chao Yan ; Korobov2 . Also,
a variational strategy using a sequence of Hylleraas-type basis functions has
been put forward in the study of the general three-body Coulomb problem for
arbitrary particle masses and charges Frolov ; Frolov2 . All of these studies
quantitatively reproduced the energy and other physical properties,
particularly of He-like atomic ions. However, no explicit approximate
expression for the ground state energy $E(Z)$ is given. Moreover, the
wavefunctions used in such calculations require of a large number of terms
(some of them up to thousands) hence hindering a clear understanding of their
physical features.

So far, the pioneering work by Hylleraas and Majorana for the variational
treatment of the three-dimensional He-like system has not only been of
hallmark importance for further refinements in the accurate treatment of
three-body Coulomb systems, but as demonstrated in TVH  at $d=3$ an Hylleraas-
type function for the ground state is one of the simplest exact solutions of
the effective theory which reproduces non-relativistic quantum electrodynamics
effects. Also, accurate knowledge of the ground-state energy allows to find
non-relativistic QED effects and, perturbatively, the relativistic QED
correction TVH . In this connection, the interested reader is kindly addressed
to the review paper Drake 2020  for a recent account of progress in assessing
the accuracy of atomic and molecular data regarding theoretical or
computational work.

We now ask how appropriate is the Hylleraas-Majorana solution in
$d-$dimensions for the study of He-like ions and, in general, of three-body
Coulomb systems. Recently TME3-d ; Willard:2018  several aspects of a general
quantum three-body system in $d$-dimensions $(d>1)$ with arbitrary masses have
been reported (see also Gu ). Needless to say, the $d-$dimensional He atom has
been analyzed in some detail by several authors Herrick -Herschbach  mainly
within the infinite nuclear mass approximation. Again, all these treatments
require of elaborate theoretical and numerical procedures which make it
difficult to follow a clear physical picture of the role of the relevant
parameters involved. Hence, it is desirable to search for compact –yet
reasonably accurate– physical ground state wavefunctions in $d$-dimensions for
these systems. This is the main focus of this work as explained below.

In the present work we aim to improve the 3D - Majorana solution (1) using a
single fully correlated function $e^{-\alpha\,Z\,(r_{1}+r_{2})-\beta\,r_{12}}$
and, mainly, to extend it to de $d$-dimensional case. Additionally, we seek
for analyticity, namely a compact analytical formula (a generalized Majorana
solution) for the ground state for any $d>1$. Our goal is three-fold: firstly,
to describe in a unified way a simple analytical method for calculating - with
a reasonable accuracy - the ground state energy of a d-dimensional non-
relativistic He-like ion with fixed nucleus. This may be considered as a
natural generalization of the 3D-Majorana solution to any dimension. Secondly,
to show that this method can be easily improved and generalized to the
$d$-dimensional case of a three-body Coulomb system with arbitrary masses and
charges. This is explicitly shown for three dimensions. Thirdly, with this
compact representation, two characteristics for $d=2,3,4,5$ and $Z\in[2,10]$
are calculated: the critical charge $Z_{c}$ and the Shannon entropy
$S_{r}^{(d)}$ (see below).

Again, it is worth pointing out here that the above mentioned generalized
Majorana formula obtained in this work has an analytical expression and
provides a very good accuracy. In particular, a novel non-liner change of
variables in $r$-space $(r_{1},r_{2},r_{12})$ is introduced that allows to
evaluate explicitly the variational energy for $d=2,4,...,2N$. To the best of
our knowledge, the corresponding 2D Majorana solution is presented for the
first time.

The structure of the paper is as follows. In Section II the general setting of
the problem is explained. For the lowest values $d=2,3,4,5$ the corresponding
concrete solutions, the generalized Majorana solutions, are derived in the
Sec. III. Agreement of those results with highly accurate numerical values
reported in the literature is presented. Somewhat unexpectedly, we also found
that the case of $d$ odd allows an exact analytic solution in terms of
elementary functions. Would this property be true in four- and five-body
atomic and molecular systems: it is interesting open question. The expansion
of the variational results obtained for large $Z$ is compared with the
celebrated perturbative expansion $1/Z$ for the ground state energy. The next
Sec. IV treats two additional important properties of the system, the Shannon
entropy and the (first) critical charge. Finally, in Sec. V we construct
explicitly at $d=3$ an improved Majorana solution by direct evaluation of the
expectation value of the energy using a more general correlated trial
function. It leads to an analytic answer for arbitrary values of the charges
and masses. While the form of the reduced Hamiltonian changes, the problem
remains 3-dimensional, leading to a rather simple analytic answer which can be
immediately generalized to higher values of $d$. This section contains
discussion of possible applications to other exotic three-body systems as
well. Atomic units $(e=1,\hbar=1)$ are used throughout unless otherwise
specified.

##  II Generalities

Assuming $r^{-1}$ Coulomb interactions, the Hamiltonian of a $d$-dimensional
Helium-like ion with static nuclei $Z>0$ takes the form:

${\cal H}\ =\ -\frac{1}{2}\,\Delta^{(d)}_{1}\ -\ \frac{1}{2}\Delta^{(d)}_{2}\
-\ \frac{Z}{r_{1}}\ -\ \frac{Z}{r_{2}}\ +\ \frac{1}{r_{12}}\ ,$ (2)

where $r_{1(2)}$ is the distance from nuclear charge $Z$ to $i$th electron,
$r_{12}$ is the inter-electronic distance, and $\Delta^{(d)}_{1(2)}$ is
$d$-dimensional Laplacian associated with $i$th electron. The eigenvalue
problem for ${\cal H}$ is defined on the configuration space
${\mathbb{R}}^{2d}$.

Since the potential depends on the Hylleraas coordinates $r_{1},r_{2}$ and
$r_{12}$ only, it seems natural to seek square integrable eigenfunctions of
(2), ${\cal H}\,\psi=E\,\psi$, in the form:

$\psi\ =\ \psi(\,r_{1},\,r_{2},\,r_{12}\,)\ .$ (3)

The ground state is of this type. It can be shown that these solutions (3)
with zero total angular momentum are described by a reduced Hamiltonian
operator ${\cal H}_{r}$. Explicitly TME3-d

${\cal H}_{r}\ =\ -\frac{1}{2}\Delta_{r}^{(3)}(r_{1},\,r_{2},\,r_{12})\ -\
\frac{Z}{r_{1}}\ -\ \frac{Z}{r_{2}}\ +\ \frac{1}{r_{12}}\ ,$ (4)

where

$\begin{split}\Delta_{r}^{(3)}\ =\
&\frac{1}{r_{1}^{d-1}}\frac{\partial}{\partial
r_{1}}\left(r_{1}^{d-1}\frac{\partial}{\partial r_{1}}\right)\ +\
\frac{1}{r_{2}^{d-1}}\frac{\partial}{\partial
r_{2}}\left(r_{2}^{d-1}\frac{\partial}{\partial r_{2}}\right)\ +\
\frac{2}{r_{12}^{d-1}}\frac{\partial}{\partial
r_{12}}\left(r_{12}^{d-1}\frac{\partial}{\partial r_{12}}\right)\\\ &\
+\frac{r_{1}^{2}+r_{12}^{2}-r_{2}^{2}}{r_{1}\,r_{12}}\,\frac{\partial}{\partial
r_{12}}\left(\frac{\partial}{\partial r_{1}}\right)\ +\
\frac{r_{1}^{2}+r_{12}^{2}-r_{1}^{2}}{r_{2}\,r_{12}}\,\frac{\partial}{\partial
r_{12}}\left(\frac{\partial}{\partial r_{2}}\right)\ .\end{split}$ (5)

The Hamiltonian (4) describes a three-dimensional particle moving in a curved
space. As mentioned above, it describes the angle-independent solutions ($\cal
S$-states) of the original Hamiltonian (2). The corresponding configuration
space is given by $\Omega\,\subset\,{\mathbb{R}}_{+}^{3}$. The ranges of the
three variables $r_{ij}$ are coupled in that they must satisfy a “triangle
condition”: their lengths must be such that they can form a triangle.

The operator (4) is essentially self-adjoint with respect to the radial
measure

$dV\ =\
\frac{2^{d}\,\pi^{d-1}}{(d-2)!}\,r_{1}\,r_{2}\,r_{12}\,S^{d-3}\,dr_{1}\,dr_{2}\,dr_{12}\qquad,\qquad\
(d>1)\ ,$ (6)

$S\ =\
\frac{1}{4}\sqrt{(r_{1}\,+\,r_{2}\,+\,r_{12})(r_{1}\,+\,r_{2}\,-\,r_{12})(r_{1}\,-\,r_{2}\,+\,r_{12})(-r_{1}\,+\,r_{2}\,+\,r_{12})}\
,$

$S$ is the area of the triangle formed by the two electrons and the charge
$Z$. The volume element (6) is greatly simplified at $d=3$,
$dV=8\,\pi^{2}\,r_{1}\,r_{2}\,r_{12}\,dr_{1}\,dr_{2}\,dr_{12}$.

In $dV=w(r_{1},\,r_{2},\,r_{12};\,d)\,dr_{1}\,dr_{2}\,dr_{12}$ the weight
function $w\propto r_{1}\,r_{2}\,r_{12}\,S^{d-3}$ for odd $d\geq 3$ becomes a
polynomial $w\rightarrow P_{N}(r_{1},\,r_{2},\,r_{12})$ of degree $N=2\,d-3$
whilst at even $d\geq 2$ it is an irrational algebraic expression.

The expectation value of the Hamiltonian (4), the energy functional, is given
by

$E_{\rm var}[\psi,\,d]\ =\
\frac{\int_{0}^{\infty}\int_{0}^{\infty}\int_{|r_{1}-r_{2}|}^{r_{1}+r_{2}}\,(\psi^{\ast}\,{\cal
H}_{r}\,\psi)\,w\,dr_{1}\,dr_{2}\,dr_{12}}{\int_{0}^{\infty}\int_{0}^{\infty}\int_{|r_{1}-r_{2}|}^{r_{1}+r_{2}}\,{\mid\psi\mid}^{2}\,w\,dr_{1}\,dr_{2}\,dr_{12}}\
.$ (7)

A considerable simplification results from going over to the so-called
perimetric coordinates used by Pekeris in his helium calculations PEKERIS .
These perimetric coordinates are defined by the linear relations

$\displaystyle\eta\ =\ -r_{1}\ +\ r_{2}\ +\ r_{12}$ (8) $\displaystyle\sigma\
=\ r_{1}\ -\ r_{2}\ +\ r_{12}$ $\displaystyle\tau\ =\ 2\,(r_{1}\ +\ r_{2}\ -\
r_{12})$ $\displaystyle dr_{1}\,dr_{2}\,dr_{12}\ =\
\frac{1}{8}d\eta\,d\sigma\,d\tau\ .$

By the above transformation the limits of the three integrals in $E_{\rm var}$
(7) become independent of each other

$E_{\rm var}[\psi,\,d]\ =\
\frac{\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\,(\psi^{\ast}\,{\cal
H}_{r}\,\psi)\,w\,d\eta\,d\sigma\,d\tau}{\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\,{\mid\psi\mid}^{2}\,w\,d\eta\,d\sigma\,d\tau}\
.$ (9)

Now, for the ground state and arbitrary $d$ let us take the celebrated
Hylleraas-type trial function

$\psi_{\small\rm Hs}\ =\ e^{-\alpha\,Z\,(r_{1}\,+\,r_{2})\,-\,\beta\,r_{12}}\
=\
e^{-\frac{\alpha\,Z}{2}(\eta+\sigma+\tau)\,-\,\frac{\beta}{2}(\eta\,+\,\sigma)}\
,$ (10)

where $\alpha>0$ and $\beta\leq 0$ are non-linear variational parameters. They
define the electron-nuclei and the electron-electron cusp parameters,
respectively. Hence, they measure the local quality of the approximate wave
function near the Coulomb singularities. The one term function (10)
incorporates the electronic correlation explicitly via the exponent
$\beta\,r_{12}$.

For actual calculations, we further introduce a new nonlinear change of
variables in the $r-$space given by

$\displaystyle u_{1}\ =\ \eta\ +\ \sigma\ =\ r_{12}\ -\
3\,(\,r_{2}\,-\,r_{1}\,)$ (11) $\displaystyle u_{2}\ =\ \eta\,\sigma\ =\
2\,(\,r_{1}\,-\,r_{2}\,)(\,r_{1}\,+\,r_{12}\,-\,r_{2}\,)$ $\displaystyle
u_{3}\ =\ \tau\ =\ 2\,(r_{1}\ +\ r_{2}\ -\ r_{12})$ $\displaystyle
dr_{1}\,dr_{2}\,dr_{12}\ =\ \frac{1}{8}d\eta\,d\sigma\,d\tau\ =\
\frac{1}{2}\frac{1}{\sqrt{u_{1}^{2}\,-\,4\,u_{2}}}du_{1}\,du_{2}\,du_{3}\ .$

It turns out that for even $d$, unlike the perimetric coordinates (8), these
$u-$variables (11) allow us to evaluate the triple integral $E_{\rm var}$ (7)
analytically in closed form. To the best of our knowledge this set of
$u$-coordinates is presented for the first time.

Note that explicit analytical formulas for the expectation value $\langle
P\rangle_{\small\psi_{\small\rm Hs}}$ of any two-variable polynomial function
$P=P(r_{1}+r_{2},r_{12})$ can be derived in a simple manner as well. To this
end, let us take the denominator in $E_{\rm var}[\psi_{\small\rm Hs},\,d]$ (9)

$\Lambda(\alpha,\,\beta)\ =\ {\rm Den}\,E_{\rm var}\ ,$ (12)

which can be evaluated exactly. From (9), we immediately obtain the relation
Calais

$\langle\,P(r_{1}+r_{2},r_{12})\,\rangle\ =\
\frac{1}{\Lambda(\alpha,\,\beta)}P\bigg{(}-\frac{1}{2\,Z}\partial_{\alpha},\,-\frac{1}{2}\partial_{\beta}\bigg{)}\,\Lambda(\alpha,\,\beta)\
,$ (13)

where
$\partial_{\alpha}\equiv\frac{\partial}{\partial\,\alpha},\,\partial_{\beta}\equiv\frac{\partial}{\partial\,\beta}$.
It implies that $E_{\rm var}$ plays the role of a generating function.

##  III Generalized ground state Majorana solutions: concrete results

In this Section, we present the generalized Majorana solutions (GMS) for the
lowest values of $d=2,3,4,5$. They are obtained from the exponential
correlated Hylleraas-type trial function $\psi_{\small\rm
Hs}=e^{-\alpha\,Z\,(r_{1}\,+\,r_{2})\,-\,\beta\,r_{12}}$ (10).

###  III.1 Case $d=3$

Let us start with the most relevant physical situation $d=3$. In this case,
the factor $w$ appearing in the functional energy $E_{\rm var}$ (9) is a cubic
polynomial in perimetric coordinates (8),
$w(d=3)\propto(\eta+\sigma)(2\,\eta+\tau)(2\,\sigma+\tau)$. We immediately
obtain the variational energy with respect to (10)

$E_{\rm var}^{(3D)}\ =\ \frac{(\beta+\alpha
Z)\left(\beta^{3}+\beta^{2}+8\alpha^{3}Z^{3}-16\alpha^{2}Z^{3}+7\alpha^{2}\beta
Z^{2}+5\alpha^{2}Z^{2}-4\alpha\beta Z^{2}+4\alpha\beta^{2}Z+4\alpha\beta
Z\right)}{\beta^{2}+8\alpha^{2}Z^{2}+5\alpha\beta Z}\ ,$ (14)

hence, it is a rational function of the non-linear parameters $\alpha$ and
$\beta$. At $\beta=0$, the analytical expression (14) reduces to

$E_{\rm var}^{(3D)}(\beta=0)\ =\ (\alpha-2)\,\alpha\,Z^{2}\ +\
\frac{5\,\alpha\,Z}{8}\ .$ (15)

At $\alpha=1,\,\beta=0$, it further simplifies to the Majorana expression (1)
with $b=0$

$E_{\rm var}^{(3D)}(\alpha=1,\,\beta=0)\ =\ -Z^{2}\ +\ \frac{5}{8}\,Z\ .$ (16)

Moreover, as a result of the minimization procedure the optimal parameters
$\alpha$ and $\beta$ turn out to be almost constant in the domain $2\leq Z\leq
10$, see Fig. 2. Thus, at $\alpha=\alpha_{\rm average}$ and $\beta=\beta_{\rm
average}$111We denote by $\alpha_{\rm average}$ the average value of the
optimal parameter $\alpha$ in the region $2\leq Z\leq 10$., we call (14)
generalized 3D Majorana solution. In particular, for the ground state of the
Helium atom ($Z=2$) the minimization of (14) gives the result

$E_{0}^{(3D)}(Z=2)\ =\ -2.889618\,a.u.$

at $\alpha=0.929044,\,\beta=-0.254746$, in complete agreement with Aquino .
Hence, a nuclear-electron cusp $\nu_{1}=-1.858$ and an electron-electron cusp
$\nu_{2}=0.2547$ . They have to be compared with the _exact_ results by
Schwartz Schwartz :

$E_{\rm exact}^{(3D)}(Z=2)\ =\
-2.903724\,a.u.\qquad,\qquad\nu_{1}^{\rm(exact)}\ =\
-2\qquad,\qquad\nu_{2}^{\rm(exact)}\ =\ \frac{1}{2}\ .$

The difference in the energy of $0.014$ a.u. $\sim 0.380$eV is the actual
error. For comparison, $k_{B}\,T\sim 0.026$ eV at room temperature. Note that
in the Hartree–Fock approximation, $\beta=0$, the error ($\sim 1.14$eV) is
three times larger. One might say that all of chemistry is buried in the
correlation energy, and that is why it is so important to get it right.

It is evident that the two-parametric trial function (10), which provides
rather accurate results, can be improved even without breaking the
aforementioned analyticity of $E_{\rm var}$. As mentioned in the Introduction
such a program has been realized at $d=3$ (see also Harris ). In order to keep
the calculations as simple as possible we choose (10) to focus on the
conceptual part of the present consideration.

####  III.1.1 $\frac{1}{Z}$ and Puiseux expansions

In this subsection, in order to estimate the quality of the 3D GMS we will
check its compatibility with well-established results found from perturbation
theory.

Firstly, it is known that at large $Z$ the ground state energy can be
estimated by the $1/Z$ perturbative expansion Kato:1980 ,

$E(Z)\ =\ -B_{0}\,Z^{2}\ +\ B_{1}\,Z\ +\ B_{2}\ +\
O\bigg{(}\frac{1}{Z}\bigg{)}\ ,$ (17)

where $B_{0}$ is the sum of energies of $2$ Hydrogenic atoms and $B_{1}$ is
the so-called electronic interaction energy. They can be calculated
analytically. In atomic units, the first three coefficients in (17) are given
by

$B_{0}\ =\ 1\ ,\qquad B_{1}\ =\ \frac{5}{8}\ =\ 0.625\ ,\qquad B_{2}\
=\,-0.15766642946915\ ,$ (18)

$B_{0,1}$ being rational numbers. Remarkably, it was proved that the expansion
(17) has a finite radius of convergence Kato:1980 .

Let us take the optimal parameters $\alpha=0.929044,\,\beta=-0.254746$
obtained at $Z=2$, they remain almost constant in the interval $2\leq Z\leq
10$. Expanding the function $E_{\rm var}^{(3D)}$ (14) in powers of $1/Z$ we
obtain

$E_{\rm var}^{(3D)}(Z)\ =\ -0.994965\,Z^{2}\ +\ 0.603247\,Z\ -\ 0.11955\ +\
O\bigg{(}\frac{1}{Z}\bigg{)}\ ,$ (19)

hence, there is a good agreement with the exact values (18).

Secondly, at small nuclear charges $Z$, Stillinger Stillinger:1966  proposed a
Puiseux expansion for the ground state energy in the form

$\begin{split}E(Z)\ =\ &E_{B}\ +\ p_{1}\left(Z-Z_{B}\right)\ +\
q_{{3}}\left(Z-Z_{B}\right)^{3/2}+p_{{2}}\left(Z-{\it Z_{B}}\right)^{2}\ +\
q_{{5}}\left(Z-Z_{B}\right)^{5/2}\\\ &\ +\
p_{{3}}\left(Z-Z_{B}\right)^{3}+q_{{7}}\left(Z-Z_{B}\right)^{7/2}\ +\
p_{{4}}\left(Z-Z_{B}\right)^{4}+\ldots\ ,\end{split}$ (20)

where $Z_{B}>0$ is a certain critical charge and $E_{B}=E(Z_{B})$. This was
confirmed quantitatively in TG:2011 -TLO:2016 . Moreover, the expansion (20)
was derived numerically in HOAT  using highly accurate values of ground state
energy in close vicinity of $Z>Z_{B}$ obtained variationally. It turns out
that $Z_{B}$ is not equal to the (first) critical charge $Z_{c}$ where the
one-electron ionization energy vanishes and, unlike the aforementioned $1/Z$
expansion, there is no a rigours proof of the convergence in the Puiseux
expansion (20). Explicitly, the first coefficients in (20) are HOAT

$Z_{B}\ =\ {0.9048539992}\ ,\qquad E_{B}\ =\ {-0.407932489}\ ,\qquad p_{1}\
=\,{-1.123475}\ ,$

$\ q_{3}\ =\,-0.197785\ ,\qquad p_{2}\ =\,-0.752842\ ,$ (21)

cf. TLO:2016 .

Now, we take the same optimal parameters $\alpha=0.929044,\,\beta=-0.254746$
obtained at $Z=2$, and expanding the function $E_{\rm var}^{(3D)}$ around
$Z=Z_{B}$ we arrive to the expression

$E_{\rm var}^{(3D)}(Z)\ =\ -\ 0.380745\ -\ 1.20635\,(Z-Z_{B})\ -\
0.984326\,{(Z-Z_{B})}^{2}\ +\ \ldots\ ,$ (22)

thus, in spite of the fact that (22) does not contain terms with fractional
degrees it does provide reasonable values for the first coefficients in front
of the terms with integer degree.

###  III.2 Case $d=2$

Now, we treat the less studied planar case $d=2$. Such a system occurs in
condensed matter physics. For the ground state we use the Hylleraas-type trial
function $\psi_{\small\rm Hs}$ (10) again.

The weight function $w$ in (9) becomes

$w(d=2)\ \propto\
\frac{(\eta+\sigma)(2\,\eta+\tau)(2\,\sigma+\tau)}{\sqrt{\eta\,\sigma\,\tau\,(2\,\eta+2\,\sigma+\tau)}}\
=\
\frac{u_{1}\,[{(u_{1}+u_{3})}^{2}\,-\,(u_{1}^{2}-4\,u_{2})]}{\sqrt{u_{2}\,u_{3}\,(2\,u_{1}+u_{3})}}\
.$

Using the $u$-variables (11) we were able to calculate the expectation value
$\langle r_{12}^{-1}\rangle_{\small\psi_{\small\rm Hs}}$ in a closed-
analytical form. The variational energy can be written as follows

$E_{\rm var}^{(2D)}\ =\
(\alpha^{2}\,Z^{2}\,-\,\beta^{2})\,\frac{A(\alpha\,Z,\,\beta,\,Z)}{B(\alpha\,Z,\,\beta,\,Z)}\
,$ (23)

where

$\displaystyle A\ =\
-2\beta^{5}(\beta+2)+8(\alpha-4)\alpha^{5}Z^{6}+\alpha^{3}\beta
Z^{4}(-9\alpha\beta-10\alpha+16\beta)+\alpha\beta^{3}Z^{2}(3\alpha\beta+14\alpha+16\beta)$
(24) $\displaystyle\qquad\ -\
3\,\alpha^{3}Z^{4}(\alpha(3\beta-2)-16\beta)\sqrt{\alpha^{2}Z^{2}-\beta^{2}}\cos^{-1}\left(\frac{\beta}{\alpha
Z}\right)\ ,$ $\displaystyle B\ =\
2\,\beta^{6}+8\,\alpha^{6}\,Z^{6}+\alpha^{4}\,\beta^{2}\,Z^{4}-11\,\alpha^{2}\,\beta^{4}\,Z^{2}-15\,\alpha^{4}\,\beta
Z^{4}\sqrt{\alpha^{2}Z^{2}-\beta^{2}}\cos^{-1}\left(\frac{\beta}{\alpha
Z}\right)\ ,$

(c.f. (14)). Therefore, unlike the previous 3D case, it is not an algebraic
function of the parameters $\alpha$ and $\beta$. To the best of our knowledge
the analytical expression (23) is presented for the first time. At $\beta=0$,
(23) reduces to

$E_{\rm var}^{(2D)}(\beta=0)\ =\ (\alpha-4)\,\alpha\,Z^{2}\ +\
\frac{3\,\pi\,\alpha\,Z}{8}\ .$ (25)

(c.f. (15),(1)). We call (23) the 2D-generalized Majorana solution. In
particular, for the ground state of the Helium atom ($Z=2$) we obtain the
energy

$E_{0}^{(2D)}(Z=2)\ =\ -11.8350\,a.u.\ ,$

at $\alpha=1.87638,\,\beta=-0.53717$. Hence, a nuclear-electron cusp
$\nu_{1}=-3.75277$ and an electron-electron cusp $\nu_{2}=0.5372$ . It has to
be compared with the _exact_ numerical result Hilico :

$E_{\rm exact}^{(2D)}(Z=2)\ =\ -11.8998\,a.u.\ .$

It implies a small relative error ($\sim 0.5\%$) in the variational energy.
Now, the denominator (12) takes the form

$\Lambda^{(2D)}(\alpha,\,\beta)\ =\ \frac{\pi\left(\sqrt{(\alpha
Z-\beta)(\beta+\alpha
Z)}\left(-2\beta^{4}+8\alpha^{4}Z^{4}+9\alpha^{2}\beta^{2}Z^{2}\right)-15\alpha^{4}\beta
Z^{4}\sec^{-1}\left(\frac{\alpha
Z}{\beta}\right)\right)}{16\alpha^{2}Z^{2}((\alpha Z-\beta)(\beta+\alpha
Z))^{7/2}}\ .$ (26)

By taking the polynomial function $P(r_{1},\,r_{2},\,r_{12})$ as the inter-
electronic distance $P=r_{12}$ and using the relation (13) we obtain

$\langle\,r_{12}\,\rangle\,=\,\frac{-4\beta^{7}-81\alpha^{6}\beta
Z^{6}+53\alpha^{4}\beta^{3}Z^{4}+32\alpha^{2}\beta^{5}Z^{2}+15\alpha^{4}Z^{4}\left(6\beta^{2}+\alpha^{2}Z^{2}\right)\sqrt{\alpha^{2}Z^{2}-\beta^{2}}\sec^{-1}\left(\frac{\alpha
Z}{\beta}\right)}{2(\alpha^{2}Z^{2}-\beta^{2})\left(2\beta^{6}+8\alpha^{6}Z^{6}+\alpha^{4}\beta^{2}Z^{4}-11\alpha^{2}\beta^{4}Z^{2}-15\alpha^{4}\beta
Z^{4}\sqrt{\alpha^{2}Z^{2}-\beta^{2}}\sec^{-1}\left(\frac{\alpha
Z}{\beta}\right)\right)}\ .$ (27)

####  III.2.1 $\frac{1}{Z}$ Expansion

In this subsection, it is worth analyzing the 2D GMS with respect to the $1/Z$
expansion. Assuming $Z\rightarrow\infty$, we immediately calculate the ground
state energy in perturbation theory in the small parameter $1/Z$,

$E^{(2D)}(Z)\ =\ -C_{0}\,Z^{2}\ +\ C_{1}\,Z\ +\ C_{2}\ +\
O\bigg{(}\frac{1}{Z}\bigg{)}\ ,$ (28)

where $C_{0}$ is the sum of energies of $2$ two-dimensional Hydrogenic atoms
and $C_{1}$ is the so-called electronic interaction energy. They can be
calculated analytically,

$C_{0}\ =\ 4\ ,\qquad C_{1}\ =\ \frac{3\,\pi}{4}\ \approx\ 2.3562\ ,$ (29)

c.f. (17), (18). Now, let us take the optimal parameters
$\alpha=1.876383,\,\beta=-0.537171$ obtained at $Z=2$ (they remain almost
constant in the interval $2\leq Z\leq 10$, see Fig. 2). Expanding the GMS
$E_{\rm var}^{(2D)}$ (23) in powers of $1/Z$ we obtain

$E_{\rm var}^{(2D)}(Z)\ =\ -3.98472\,Z^{2}\ +\ 2.28879\,Z\ -\ 0.490893\ +\
O\bigg{(}\frac{1}{Z}\bigg{)}\ ,$ (30)

c.f. (19). Not surprisingly, a good agreement with the exact values occurs.

###  III.3 Case $d=4$

Now, we move to the case $d=4$. The variational energy with respect to (10) is
given by

$E_{\rm var}^{(4D)}\ =\ (\alpha^{2}\,Z^{2}\,-\,\beta^{2})\,\frac{{\cal
A}(\alpha\,Z,\,\beta,\,Z)}{{\cal B}(\alpha\,Z,\,\beta,\,Z)}\ ,$ (31)

where

$\displaystyle{\cal A}\ =\
2q_{1}[16\beta^{7}(3\beta+2)+256\alpha^{7}(4-3\alpha)Z^{8}+13\alpha^{5}\beta
Z^{6}(-303\alpha\beta+102\alpha+608\beta)$ (32) $\displaystyle\qquad\ +\
10\alpha^{3}\beta^{3}Z^{4}(15\alpha\beta+74\alpha+128\beta)-8\alpha\beta^{5}Z^{2}(27\alpha\beta+26\alpha+16\beta)]$
$\displaystyle+105\alpha^{4}Z^{5}\left(8\beta^{2}(3\alpha\beta-2\alpha-8\beta)+\alpha^{2}Z^{2}(21\alpha\beta-2\alpha-32\beta)\right)\left(\pi\alpha^{2}Z^{2}-2\csc^{-1}\left(\frac{\alpha
Z}{\beta}\right)\right)\ ,$ $\displaystyle{\cal B}\ =\ 945\alpha^{5}\beta
Z^{5}\left(8\beta^{2}+3\alpha^{2}Z^{2}\right)\left(\pi\alpha^{2}Z^{2}-2\csc^{-1}\left(\frac{\alpha
Z}{\beta}\right)\right)$ $\displaystyle\qquad\
-6\,q_{1}\,\left(16\beta^{8}+256\alpha^{8}Z^{8}+2639\alpha^{6}\beta^{2}Z^{6}+690\alpha^{4}\beta^{4}Z^{4}-136\alpha^{2}\beta^{6}Z^{2}\right)\
,$

here $q_{1}=\sqrt{1-\frac{\beta^{2}}{\alpha^{2}Z^{2}}}$. Hence, like for
$d=2$, it is not an algebraic function of the variational parameters. As
mentioned in the Introduction, to evaluate the energy functional for even $d$
we employed the $u$-variables (11) instead of the perimetric ones. At
$\beta=0$, the analytical expression (31) reduces to

$E_{\rm var}^{(4D)}(\beta=0)\ =\ 1\ -\ \frac{4}{3\,\alpha}\ +\
\frac{35\,\pi\,\alpha\,Z}{256}\ .$ (33)

(c.f. (1),(15),(25)). Thus, we call (31) the 4D-generalized Majorana solution.
Accordingly, for the ground state of the Helium atom ($Z=2$) we arrive to the
value

$E_{0}^{(4D)}(Z=2)\ =\ -1.26809\,a.u.\ ,$

at $\alpha=0.6160175,\,\beta=-0.1650859$. Hence, a nuclear-electron cusp
$\nu_{1}=-1.232035$ and an electron-electron cusp $\nu_{2}=0.1651$ . It has to
be compared with the _exact_ numerical result Duan :

$E_{\rm exact}^{(4D)}(Z=2)\ =\ -1.27364\,a.u.\ .$

It implies a small relative error ($\sim 0.43\%$) in the variational energy.

####  III.3.1 $\frac{1}{Z}$ Expansion

Assuming $Z\rightarrow\infty$, let us determine the ground state energy in
perturbation theory in the small parameter $1/Z$,

$E^{(4D)}(Z)\ =\ -D_{0}\,Z^{2}\ +\ D_{1}\,Z\ +\ D_{2}\ +\
O\bigg{(}\frac{1}{Z}\bigg{)}\ ,$ (34)

where $D_{0}$ is the sum of energies of $2$ four-dimensional Hydrogenic atoms
and $D_{1}$ is the so-called electronic interaction energy. They also can be
found exactly,

$D_{0}\ =\ \frac{4}{9}\ \approx\ 0.444\ ,\qquad D_{1}\ =\ \frac{35\,\pi}{384}\
\approx\ 0.28634\ ,$ (35)

c.f. (29), (18). Taking the optimal parameters
$\alpha=0.6160175,\,\beta=-0.1650859$ obtained at $Z=2$ and expanding for
large $Z\rightarrow\infty$ the GMS $E_{\rm var}^{(2D)}$ (31) in powers of
$1/Z$, we arrive to the expression

$E_{\rm var}^{(4D)}(Z)\ =\ -0.441879\,Z^{2}\ +\ 0.275362\,Z\ -\ 0.0525401\ +\
O\bigg{(}\frac{1}{Z}\bigg{)}\ ,$ (36)

thus, in excellent agreement with the exact values.

###  III.4 Case $d=5$

For the case $d=5$, the weight function $w$ appearing in $E_{\rm var}$ (9)
becomes
$w(d=5)\propto\eta\,\sigma\,\tau\,(\eta+\sigma)(2\,\eta+\tau)(2\,\sigma+\tau)(2\,\eta+2\sigma+\tau)$.
The variational energy with respect to (10) reads

$E_{\rm var}^{(5D)}\ =\
(\,\beta\,+\,\alpha\,Z)\,\frac{\tilde{A}(\alpha\,Z,\,\beta,\,Z)}{\tilde{B}(\alpha\,Z,\,\beta,\,Z)}\
,$ (37)

where

$\displaystyle\tilde{A}\ =\ \alpha^{3}\,Z^{3}\,(\,64(\alpha-1)Z\,+\,21)\ +\
\alpha^{2}\,\beta\,Z^{2}\,(14(5\,\alpha\,-\,2)Z\,+\,19)$ (38)
$\displaystyle\qquad\ +\ \beta^{3}\,(14\,\alpha\,Z\,+\,1)\ +\
\alpha\,\beta^{2}\,Z\,((42\,\alpha-4)Z\,+\,7)\ +\ 2\,\beta^{4}\ ,$
$\displaystyle\tilde{B}\ =2\,\left(\,32\,\alpha^{3}\,Z^{3}\ +\
25\,\alpha^{2}\,\beta\,Z^{2}\ +\ 8\,\alpha\,\beta^{2}\,Z\ +\
\beta^{3}\,\right)\ \ ,$

(c.f. (14)) thus, it is a polynomial function of the variational parameters.
At $\beta=0$, the energy (37) reduces to

$E_{\rm var}^{(5D)}(\beta=0)\ =\
\frac{1}{64}\alpha\,Z^{2}\,\left(\,64\,\alpha\,Z^{2}\ -\ 64\,Z\ +\
21\,\right)\ .$ (39)

(c.f. (1),(15)). The expression (37) can be called the 5D-GMS.

Eventually, for the Helium atom ($Z=2$) we obtain the ground state energy

$E_{0}^{(5D)}(Z=2)\ =\ -0.7077\,a.u.$

at $\alpha=0.460444,\,\beta=-0.121678$. Hence, a nuclear-electron cusp
$\nu_{1}=-0.920888$ and an electron-electron cusp $\nu_{2}=0.1217$ . It has to
be compared with the _exact_ numerical result Duan :

$E_{\rm exact}^{(5D)}(Z=2)\ =\ -0.71050\,a.u.\ .$

In this case, the relative error in the variational energy continues to be
small $\sim 0.43\%$.

In Table 1, for the lowest values of $d=2,3,4,5$ we display the ground state
energy $E_{0}=E_{0}(Z)$ for a $d$-dimensional helium-like ion in the infinite
nuclear mass approximation. The results were obtained using the 2-parametric
Hylleraas-type trial function (10). Due to the analyticity of the formalism,
one can treat $Z$ as a continuous variable in the non-relativistic domain
$2\leq Z\leq 10$. The corresponding plots are shown in Figure 1. The
associated variational parameters $\alpha(Z)$, $\beta(Z)$ are presented in
Figure 2. They are smooth monotonic functions of the nuclear charge $Z$.

Figure 1: Ground state energy: $-E_{0}$ vs. $Z$ for $1\leq Z\leq 10$ in static
approximation (infinite nuclear mass case). The variational energy $E_{0}$ is
calculated with the 2-parametric Hylleraas-type function (10). The energy is
in atomic units (a.u.), (color online).

Figure 2: Ground state energy: non-linear parameters $\alpha$ vs $Z$ (left)
and $\beta$ vs $Z$ (right) for $2\leq Z\leq 10$ in static approximation
(infinite nuclear mass case). These parameters define the 2-parametric
Hylleraas-type trial function (10), (color online).

Table 1: The numerical calculation for the ground state energy $E_{0}$ in a $d$-dimensional helium-like ion in the infinite nuclear mass approximation. The expectation value $\langle r_{12}\rangle$ also calculated from the 2-parametric Hylleraas-type trial function (10). Numerical results presented in atomic units. | $E_{0}\,(a.u.)$ | $\langle r_{12}\rangle\,(a.u.)$
---|---|---
$Z$ |  $d=2$ |  $d=3$ |  $d=4$ |  $d=5$ |  $d=2$ |  $d=3$ |  $d=4$ |  $d=5$
2 | -11.8350 | -2.88961 | -1.26809 | -0.707704 | 0.4668 | 1.385 | 2.754 | 4.573
3 | -29.4889 | -7.26681 | -3.20486 | -1.79411 | 0.2869 | 0.8515 | 1.692 | 2.808
4 | -55.1377 | -13.6429 | -6.03008 | -3.38029 | 0.2067 | 0.6137 | 1.219 | 2.024
5 | -88.7845 | -22.0185 | -9.74400 | -5.46636 | 0.1615 | 0.4795 | 0.9530 | 1.581
6 | -130.430 | -32.3939 | -14.3467 | -8.05240 | 0.1325 | 0.3934 | 0.7819 | 1.297
7 | -180.075 | -44.7692 | -19.8382 | -11.1384 | 0.1123 | 0.3335 | 0.6629 | 1.100
8 | -237.720 | -59.1445 | -26.2187 | -14.7243 | 0.0974 | 0.2894 | 0.5753 | 0.9548
9 | -303.365 | -75.5197 | -33.4879 | -18.8103 | 0.0860 | 0.2556 | 0.5081 | 0.8433
10 | -377.009 | -93.8948 | -41.6461 | -23.3963 | 0.0770 | 0.2289 | 0.4549 | 0.7551

##  IV The Shannon Entropy and the (first) critical charge $Z_{c}$

So far we have discussed the ability of the GMS to account for reasonably
accurate values of the ground-state energy of He-like systems with clamped
nuclei in $d=2,3,4$ and $5$ dimensions. In this context, it is worth analyzing
the behavior of another two relevant properties derived from the electronic
density and nuclear charge, namely the Shannon entropy and the (first)
critical charge $Z$.

###  IV.1 The Shannon Entropy

In many-electron systems the Shannon entropy Shannon  has been used as a
measure of the uncertainty in the position of one electron, which is related
with the degree of localization-delocalization of the particle. When the
Shannon entropy is small the uncertainty in the position of the particle is
also small, and the particle is more localized and _vice-versa_ , see Esta
-Aquino3  an references therein. In the present study the Shannon entropy will
be used as a measure of the localization (delocalization) of the particle.

More precisely, we are to determine the Shannon entropy within the formalism
developed in previous sections. For the ground state of a $d$-dimensional
Helium-like ion we shall compute the Shannon entropy in the Hylleraas
coordinates $r_{1},r_{2}$ and $r_{12}$ using the optimal trial function (10).

A $d$-dimensional two electron system is completely described by a wave
function of the form $\Psi({\bf r}_{1},{\bf r}_{2},\sigma_{1},\sigma_{2})$,
where ${\bf r}=(x_{1},x_{2},\ldots,x_{d})$. The main quantity involved in the
Shannon entropy is the single-particle density $\rho^{(d)}({\bf r})$, defined
as:

$\rho^{(d)}({\bf r}_{1})=\sum_{\sigma_{i}=-1/2}^{+1/2}\int\left|\Psi({\bf
r}_{1},{\bf r}_{2},\sigma_{1},\sigma_{2})\right|^{2}\times d{\bf r}_{2}\ ,$
(40)

where $\sigma_{i}$ represents the spin coordinate of the i$th$ particle. The
wave function is normalized and it is antisymetric under the exchange of the
particles. The single particle density $\rho^{(d)}({\bf r})$ is also
normalized to the unity.

Since the ground state function of a Helium-like ion solely depends on the
Hylleraas coordinates, the single-particle density (40) can be written in the
form

$\rho^{(d)}(r_{1})\ =\
\frac{1}{s_{d-1}(r_{1})}\int_{0}^{\infty}\int_{|r_{1}-r_{2}|}^{r_{1}+r_{2}}\,|\Psi(r_{1},r_{2},r_{12})|^{2}\,dR\
,$ (41)

where $s_{d-1}(r)=\omega_{d}r^{d-1}$, $\omega_{d}=2\pi^{d/2}/\Gamma(d/2)$ is
the area of the surface of a unit sphere in $d$-dimensions, and

$dR\ =\
\frac{2^{d}\,\pi^{d-1}}{(d-2)!}\,r_{2}\,r_{12}\,S^{d-3}\,dr_{2}\,dr_{12}\ .$

In (41), the function $\Psi(r_{1},r_{2},r_{12})$ is taken from (10).
Accordingly, the Shannon entropy is given by

$S_{r}^{(d)}\ =\
-\int_{0}^{\infty}\rho^{(d)}(r)\,\ln\rho^{(d)}(r)\,\omega_{d}\,r^{d-1}\,dr\ .$
(42)

In three dimensional space $(d=3)$ the single particle density and the Shannon
entropy reduce to the well-known expressions

$\rho(r_{1})\ =\
\frac{2\,\pi}{r_{1}}\int_{0}^{\infty}\int_{|r_{1}-r_{2}|}^{r_{1}+r_{2}}\,|\Psi(r_{1},r_{2},r_{12})|^{2}\,r_{2}\,r_{12}\,dr_{2}\,dr_{12}\
,$ (43)

and

$S_{r}\ =\ -4\,\pi\,\int_{0}^{\infty}\,\rho(r)\,\ln{\rho(r)}\,r^{2}\,dr\ .$
(44)

Table 2 displays the Shannon entropy as a function of the nuclear charge
$Z=2,\dots,10$ for $d=2,3,4$ and $5$ (see also Figure 3). Two important
features appear in the behavior of the Shannon entropy: $i$) for each fixed
value of $d$, $S^{(d)}_{r}$ is a strictly decreasing function of $Z$ and,
$ii$) in agreement with the results for the $d$-dimensional hydrogen atom
YAD:2020 , $S^{(d)}_{r}$ is a strictly increasing function of $d$. Column 4 of
Table 2 shows the results of the Shannon entropy for $d=3$ calculated in
LH:2015  (with 444 basis functions, label $a$) and NA:2020  (with a three-
parameter wave function, label $b$). For $Z=2,3$ the relative error is $\sim
3\%$ and for $Z=4,\dots,10$ the relative error decreases from $\sim 10\%$ to
$\sim 0.4\%$

$Z\quad$ |  $d=2$ |  $d=3$ |  $d=4$ |  $d=5$
---|---|---|---|---
2 | 0.0425 |  2.6194 2.7051${}^{\,a}$ | 5.5974 | 8.9218
3 | -0.8952 |  1.2090 1.2553${}^{\,a}$ | 3.7135 | 6.5642
4 | -1.5318 |  0.2524 0.2824${}^{\,b}$ | 2.4368 | 4.9673
5 | -2.0142 |  -0.4719 -0.4489${}^{\,b}$ | 1.4703 | 3.7585
6 | -2.4027 |  -1.0551 -1.0364${}^{\,b}$ | 0.6923 | 2.7857
7 | -2.7278 |  -1.5432 -1.5274${}^{\,b}$ | 0.0413 | 1.9717
8 | -3.0074 |  -1.9629 -1.9493${}^{\,b}$ | -0.5185 | 1.2717
9 | -3.2527 |  -2.3310 -2.3190${}^{\,b}$ | -1.0095 | 0.6579
10 | -3.4712 |  -2.6588 -2.6482${}^{\,b}$ | -1.4468 | 0.1112
Table 2: Shannon entropy $S_{r}^{(d)}$ for two-electron systems with nuclear
charge $Z$ in a $d$-dimensional space.

The Shannon entropy $S_{r}^{(d)}(Z)$ for $d=3,4$ and $5$ (Table 2) can be
obtained from the results for $d=2$, namely $S_{r}^{(2)}(Z)$, with the
empirical interpolation

$S_{r}^{(d)}(Z)\ =\ S_{0}^{(d)}\ +\ \frac{d}{2}S_{r}^{(2)}(Z)\ +\
\frac{(d-1)(d-2)+2}{200\,Z},\hskip 28.45274ptd=3,4,5,$ (45)

where $S_{0}^{(3)}=2.5455$, $S_{0}^{(4)}=5.4919$, $S_{0}^{(5)}=8.7805$ and $Z$
being the nuclear charge. The absolute error in the Shannon entropy calculated
with (45) is $\lesssim 10^{-3}$ compared to the numerical results of Table 2.

Figure 3: Shannon entropy $S_{r}^{(d)}$ for two-electron systems as a function
of the nuclear charge $Z$ for $d=2,3,4$ and $5$ dimensions. Numerical values
marked by bullets, the lines are a guide to the eye.

###  IV.2 The (first) critical charge $Z_{c}$

Considering the nuclear charge as a continuous parameter in the Helium-like
sequence, an important characteristic of the system is the so call (first)
critical charge $Z_{c}$ above of which, $Z>Z_{c}$, the system gets bound.
Explicitly, the charge $Z_{c}$ is defined by the relation

$E_{\rm var}^{(d)}(Z_{c})\ +\
\frac{1}{2}\frac{Z_{c}^{2}}{{(1\,+\,[d-3]/2)}^{2}}\ =\ 0\ ,$ (46)

where $-\frac{1}{2}\frac{Z_{c}^{2}}{{(1\,+\,[d-3]/2)}^{2}}$ corresponds to
ground state energy of the Hydrogen atom in $d-$dimensions.

The aforementioned generalized Majorana formulas (14), (23), (31) and (37)
give a rather accurate value of $Z_{c}$. Our variational calculations using
(10) provide the following values

$Z_{c}^{(d=2)}\ \approx\ 0.8931\ ;\,\quad Z_{c}^{(d=3)}\ \approx\ 0.9791\
;\,\quad Z_{c}^{(d=4)}\ \approx\ 1.023\ ;\,\quad Z_{c}^{(d=5)}\ \approx\
1.049\ .$ (47)

In particular, at $d=3$ the relative error with respect to the exact numerical
resultEstienne -HOAT

$\quad Z_{c,{\rm exact}}^{(d=3)}\ =\ 0.91102822\ ,$

is $\approx 0.07\%$ . Below, we will show that the addition of a certain
variational parameter in (10) does not break the analyticity (the functional
of energy can be evaluated exactly) and one can easily check that it makes the
latter relative error $\approx 0.07\%$ decrease up to $\approx 0.016$.

##  V Three-body Coulomb system: finite mass effects

In this Section, we will study in full generality the finite mass effects for
a $d-$dimensional three-body Coulomb system of arbitrary charges. The study is
restricted to the ground state of the system. To this end, we also employ the
variational method with a more general correlated Hylleraas-type trial
function.

In the case of 3 particles with arbitrary masses ($m_{1},m_{2},M$) and charges
($e_{1},e_{2},Z$), respectively, the generalization of the operator ${\cal
H}_{r}$ (4) is given by TME3-d

${\cal H}_{m,r}\ =\ -\frac{1}{2}\Delta_{m,r}^{(3)}(r_{1},\,r_{2},\,r_{12})\ +\
\frac{e_{1}\,Z}{r_{1}}\ +\ \frac{e_{2}\,Z}{r_{2}}\ +\
\frac{e_{1}\,e_{2}}{r_{12}}\ ,$ (48)

where

$r_{13}\ \equiv\ r_{1}\quad,\qquad r_{23}\ \equiv\ r_{2}\ ,$

and

$\begin{split}\Delta_{m,r}^{(3)}\ =&\
\frac{m_{1}+M}{m_{1}\,M}\frac{1}{r_{1}^{d-1}}\frac{\partial}{\partial
r_{1}}\left(r_{1}^{d-1}\frac{\partial}{\partial r_{1}}\right)\ +\
\frac{m_{2}+M}{m_{2}\,M}\frac{1}{r_{2}^{d-1}}\frac{\partial}{\partial
r_{2}}\left(r_{2}^{d-1}\frac{\partial}{\partial r_{2}}\right)\\\ &+\
\frac{m_{1}+m_{2}}{m_{1}\,m_{2}}\frac{1}{r_{12}^{d-1}}\frac{\partial}{\partial
r_{12}}\left(r_{12}^{d-1}\frac{\partial}{\partial r_{12}}\right)+\
\frac{r_{1}^{2}+r_{12}^{2}-r_{2}^{2}}{m_{1}\,r_{1}\,r_{12}}\,\frac{\partial}{\partial
r_{12}}\left(\frac{\partial}{\partial r_{1}}\right)\\\ &+\
\frac{r_{1}^{2}+r_{12}^{2}-r_{1}^{2}}{m_{2}\,r_{2}\,r_{12}}\,\frac{\partial}{\partial
r_{12}}\left(\frac{\partial}{\partial r_{2}}\right)+\
\frac{r_{1}^{2}+r_{2}^{2}-r_{12}^{2}}{M\,r_{1}\,r_{2}}\,\frac{\partial}{\partial
r_{1}}\left(\frac{\partial}{\partial r_{2}}\right)\ .\end{split}$ (49)

Thus, the problem is still 3-dimensional and corresponds to a three-
dimensional particle moving in a curved space. Putting $m_{1}=m_{2}=1$,
$M\rightarrow\infty$, $e_{1}=e_{2}=-1$, the reduced Hamiltonian ${\cal
H}_{m,r}$ (48) becomes the static-nuclei Hamiltonian ${\cal H}_{r}$ (4).

The operator (48) is essentially self-adjoint with respect to the same radial
measure $dV$ (6). Now, just as for the ground state we take the Hylleraas-type
trial function

$\psi_{\rm m}\ =\ (1\ +\
{\hat{P}}_{12})\,e^{-\alpha_{1}\,Z\,\,r_{13}\,-\,\alpha_{2}\,Z\,r_{23}\,-\,\beta\,r_{12}}\
,$ (50)

where $\alpha_{1(2)}$ and $\beta$ are variational parameters such that
$\psi_{\rm m}$ is square integrable ($\cal S$-bound states). The factor
$1+{\hat{P}}_{12}$ in (50) accounts for the spatial symmetric combination
associated with singlet states. In perimetric coordinates

$\psi_{\rm m}\ =\ (1\ +\
{\hat{P}}_{12})\,\exp\left(-\frac{1}{4}\,\alpha_{1}\,Z\,(2\,\sigma+\tau)\,-\,\frac{1}{4}\,\alpha_{2}\,Z\,(2\,\eta+\tau)\,-\,\frac{1}{2}\,\beta\,(\eta+\sigma)\right)\
.$ (51)

It is worth mentioning that explicit analytical formulas for the expectation
value $\langle Q\rangle_{\small\psi_{\small\rm m}}$, for any multi-variable
polynomial function $Q=Q(r_{1},r_{2},r_{12})$, can be obtained by simple
differentiation of the denominator Den$E_{\rm var}[\psi_{\small\rm m}]$ in (9)
which is a function of $\alpha_{1(2)}$ and $\beta$ only. From (9), we
immediately obtain the relation

$\langle\,Q(r_{1},r_{2},r_{12})\,\rangle\ =\ \frac{1}{\text{Den}E_{\rm
var}}Q\bigg{(}-\frac{1}{2\,Z}\partial_{\alpha_{1}},\,-\frac{1}{2\,Z}\partial_{\alpha_{2}},\,-\frac{1}{2}\partial_{\beta}\bigg{)}\,\text{Den}E_{\rm
var}\ ,$ (52)

where
$\partial_{\alpha_{1(2)}}\equiv\frac{\partial}{\partial\,\alpha_{1(2)}},\,\partial_{\beta}\equiv\frac{\partial}{\partial\,\beta}$.
It implies that $E_{\rm var}$ plays the role of a generating function. The
basic object appearing in the variational calculations is of the form

${\cal I}(a,\,b,\,c)\ =\
\int_{0}^{\infty}\int_{0}^{\infty}\int_{|r_{1}-r_{2}|}^{r_{1}+r_{2}}\,e^{-a\,r_{13}\,-\,b\,r_{23}\,-\,c\,r_{12}}\,dr_{1}\,dr_{2}\,dr_{12}\
=\ \frac{16}{(a+b)(a+c)(b+c)}\ ,$ (53)

$a+b>0,a+c>0,b+c>0$. Hence, from (53) we arrive to the useful expression

$\displaystyle\langle\,r_{1}^{n}\,r_{2}^{m}\,r_{12}^{k}\,\rangle$
$\displaystyle\equiv\
\int_{0}^{\infty}\int_{0}^{\infty}\int_{|r_{1}-r_{2}|}^{r_{1}+r_{2}}\,e^{-a\,r_{13}\,-\,b\,r_{23}\,-\,c\,r_{12}}\,r_{1}^{n}\,r_{2}^{m}\,r_{12}^{k}\,dr_{1}\,dr_{2}\,dr_{12}$
(54) $\displaystyle\ =\
(-1)^{n+m+k}\,\partial^{n}_{a}\,\partial_{b}^{m}\,\partial_{c}^{k}\,{\cal
I}(a,\,b,\,c)\ ,$

here $\partial^{n}_{x}\equiv\frac{\partial^{n}}{\partial\,x^{n}}$.

###  V.1 Case $d=3$: concrete results

Here we consider the three-dimensional ($d=3$) case in detail. The variational
energy $E_{\rm m,var}^{(3D)}=E_{\rm var}^{(3D)}[\psi_{\rm m}]$ is a rational
function of the three parameters $\alpha_{1(2)}$ and $\beta$. In general, it
takes the form

$E_{\rm m,var}^{(3D)}\ =\ \frac{P_{10}(Z)}{P_{8}(Z)}\quad,\qquad
P_{N}(Z)=\sum_{i=0}^{N}c_{i}^{(N)}\,Z^{i}\ ,$ (55)

where $P_{N}$ is a polynomial of degree $N$ in $Z$ with constant coefficients
$c_{i}^{(N)}=c_{i}^{(N)}(\alpha_{1},\alpha_{2},\beta,m_{1},m_{2},M,e_{1},e_{2})$.
The explicit form of these coefficients, in the case
$m_{1}=m_{2}=m,\,e_{1}=e_{2}=e$, is presented in the Appendix.

In particular, for the Helium atom ($m_{1}=m_{2}=1\,$,
$e_{1}=e_{2}=-1$,$\,Z=2$) with finite nuclear mass $M=7294.261824\,a.u.$,
minimization of $E_{\rm m,var}^{(3D)}$ with $\alpha_{1}=\alpha_{2}$ gives the
value

$E_{\rm m,varI}^{(3D)}(Z=2)\ =\ -2.8891879\,a.u.\ ,$

at $\alpha_{1}=\alpha_{2}=0.92887416,\,\beta=-0.2546058$. Releasing the
condition $\alpha_{1}=\alpha_{2}$, the variational energy turns out to be

$E_{\rm m,varII}^{(3D)}(Z=2)\ =\ -2.8991105\,a.u.\ ,$

at $\alpha_{1}=1.1031235,\,\alpha_{2}=0.72010632,\,\beta=-0.207181825$. They
have to be compared with the _exact_ result Duan :

$E_{\rm m,exact}^{(3D)}(Z=2)\ =\ -2.9033045\,a.u.\ .$

Since $E_{\rm m,var}^{(3D)}$ admits an analytical expression, it is
straightforward to study different systems. In particular, in Table 3 we show
the comparison of the ground-state energy obtained in the present study with
the results calculated in Korobov  where a large number ($\sim 1000$) of basis
functions were employed.

$\begin{array}[]{|c||c|c|c|c|c|}\hline\cr{\rm System}&\hskip
14.22636ptE_{0}\,(a.u.)&\hskip 14.22636ptE_{0}^{(a)}\,(a.u.)&\hskip
14.22636pt\alpha_{1}&\hskip 14.22636pt\alpha_{2}&\hskip 14.22636pt\beta\\\
\hline\cr
e^{-}\,e^{-}\,e^{+}&-0.256692&-0.262005&0.520138&0.147915&-0.005991\\\
\hline\cr H_{2}^{+}&-0.592720&-0.597139&0.333979&0.786127&0.445961\\\
\hline\cr H^{-}&-0.523557&-0.527751&1.0743677&0.4831917&-0.146303\\\
\hline\cr\end{array}$ Table 3: Comparison of the ground-state energy of the
positive hydrogen molecular ion and the system $e^{-}\,e^{-}\,e^{+}$ obtained
in this work ($E_{0}$) with the theoretical calculations ($E_{0}^{(a)}$)
presented in Korobov . Similarly, for the ion $H^{-}$ we compare our result
with those presented in Korobov2 .

In Table 4, we display the ground state energy $E_{\rm m,var}^{(3D)}\equiv
E_{0}$ of the 3D Helium atom and its isoelectronic ions for $Z=2,\ldots,10$.
Our variational results are compared with the highly accurate values ($\sim
35$ significant digits) obtained in Nakashima . The corresponding relative
error is presented as well. At $Z=2$, this error turns out to be $\sim 0.001$
and it decreases monotonically up to $0.00004$ at $Z=10$. It exhibits the high
quality of the Hylleraas trial function (50).

Finally, in Fig. 4 we present the corresponding comparison between the finite
and infinite nuclear mass cases within the variational approach.

$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr\hskip 8.5359ptZ&\hskip
8.5359ptE_{0}\,(a.u.)&\hskip 8.5359ptE_{0,{\rm exact}}\,(a.u.)&\hskip
11.38092pt\Delta\,E&\hskip 11.38092pt\alpha_{1}&\hskip
11.38092pt\alpha_{2}&\hskip 11.38092pt\beta\\\ \hline\cr\hline\cr
2&-2.89911&-2.903304&0.001444&1.103123&0.720106&-0.207182\\\
3&-7.27511&-7.279321&0.000578&1.099262&0.787267&-0.220592\\\
4&-13.65050&-13.654709&0.000308&1.093341&0.822938&-0.226777\\\
5&-22.02564&-22.029846&0.000191&1.087862&0.845780&-0.230357\\\
6&-32.40054&-32.404733&0.000129&1.083099&0.861923&-0.232648\\\
7&-44.77546&-44.779658&0.000094&1.078982&0.874092&-0.234331\\\
8&-59.15034&-59.154533&0.000071&1.075412&0.883655&-0.235556\\\
9&-75.52531&-75.529499&0.000055&1.072286&0.891420&-0.236509\\\
10&-93.90001&-93.904195&0.000044&1.069102&0.898483&-0.238653\\\
\hline\cr\end{array}$ Table 4: The ground state energy $E_{0}$ of 3D Helium
atom and its isoelectronic ions for $Z=2,3,\ldots,10$ with finite nuclear
mass. The values of masses were taken from (Nakashima ). The second column
represents the variational energies from the 3-parametric trial function (50).
The third column corresponds to the exact numerical values (see TVH ;
Nakashima ). The relative error $\Delta\,E=\frac{|E_{0,{\rm
exact}}-E_{0}|}{|E_{0,{\rm exact}}|}$ is displayed as well. The last three
columns show the optimal variational parameters in (50). They are smooth,
slow-changing functions of the nuclear charge $Z$. Figure 4: The ground state
energy of 3D Helium atom and its isoelectronic ions for $Z=2,3,\ldots,10$. The
relative difference $\delta E_{m}=\frac{|E_{0,\rm finite}-E_{0,\rm
infinite}|}{|E_{0,\rm finite}|}$ for $2\leq Z\leq 10$. The energies were
calculated from the 3-parametric Hylleraas-type function (50).

##  VI Conclusions

Summarizing, for Helium-like ions in $d-$dimensions an approximate compact
analytical expression for the ground state energy of the system is constructed
in the non-relativistic domain $Z\leq 10$ for the lowest values $d=2,3,4,5$.
We call them generalized Majorana solutions. The basic building block is a
Hylleraas-type trial function given by a 2-parametric correlated exponential
function in the Hylleraas variables ($r_{1},r_{2},r_{12}$). In particular, by
pure algebraic means these generalized Majorana solutions allow us to
calculate the (first) critical charge with reasonable accuracy and they
provide a good estimation of the leading terms in the celebrated $\frac{1}{Z}$
expansion. At $d=3$, a similar situation occurs for the Puiseux expansion.

Especially, it was demonstrated that for odd $d\geq 3$ the Majorana solution
is given by a rational algebraic function of the $Z$ whilst for even $d\geq 2$
it turns out to be a more complicated non-algebraic expression. In the latter
case it is quite a non-trivial task due to the elaborate form of the Jacobian
in the functional energy. To the authors knowledge, the corresponding 2D
Majorana solution is reported for the first time. The key element was the
introduction, in the 3-dimensional $r-$space, of a nonlinear change of
variables.

Originally, in the computations reported here the nuclear charge was assumed
to be of infinite mass. This assumption, though unnecessary, was adopted to
facilitate the computations and the comparison with earlier results by other
authors. At $d=3$, an improved 3-parametric correlated Hylleraas-type trial
was used to compute the finite mass effects in the generalized Majorana
solution for a three-body Coulomb system with arbitrary charges and masses.
Similar to the infinite nuclear mass caseHOAT , it leads to surprisingly
accurate values where the relative error smoothly decreases from $10^{-3}$ at
$Z=2$ up to $10^{-5}$ at $Z=10$. In the present study concrete results, beyond
the Born-Oppenheimer approximation, for the systems $e^{-}\,e^{-}\,e^{+}$,
$H_{2}^{+}$ and $H^{-}$ were calculated explicitly.

We also showed that the Shannon entropy for Helium-like systems exhibits the
following behavior: i) for a fixed value of $d$, the Shannon entropy is a
monotonic decreasing function of the nuclear charge $Z$, this is explained by
the fact that the larger the $Z$, the greater the Coulomb attraction between
the electrons and the nucleus is, hence, the electrons are located at a
smaller distance from the nucleus (localization), and ii) for a fixed value of
the nuclear charge $Z$ the Shannon entropy grows with dimensionality, a
similar situation occurs for the hydrogen atom in $d$-dimensions.

Finally, when the hydrogen-like atoms are confined inside an impenetrable
spherical box of radius $R_{c}$, the relativistic effects become more
significant, especially in the strong confinement regime. It is important to
know the value of relativistic corrections such as the kinetic energy
correction, spin-orbit coupling and the one-body and two-body Darwin terms, as
a function of the confinement radius $R_{c}$, nuclear charge $Z$ and the space
dimensionality $d$.

Quantum entanglement in the ground state of the helium atom has been studied
in three dimensions by means of von Neumann entropy and linear entropy. It is
interesting to explore entanglement in the helium atom in $d$-dimensions.

Acknowledgments

One of the authors (HOP) thanks the support through the Programa Especial de
Apoyo a la Investigación 2019, Universidad Autónoma Metropolitana, (I30). AMER
thanks the support through the Programa Especial de Apoyo a la Investigación
2021, UAM-I.

##  VII DATA AVAILABILITY

The authors confirm that the data supporting the findings of this study are
available within the article.

##  Appendix A Coefficients $c_{i}^{(N)}$

For $m_{1}=m_{2}=m,\,e_{1}=e_{2}=e$, the coefficients $c_{i}^{(N)}$ appearing
in $E_{\rm m,var}^{(3D)}$ (55). For the numerator
$P_{10}(Z)=\sum_{i=0}^{10}c_{i}^{(10)}\,Z^{i}$ we obtain

$\displaystyle c_{10}^{(10)}\ =\
\alpha_{s}^{2}\,\bigg{[}\alpha_{s}^{6}\left(\alpha_{s}\left(2\,e\,m\,M+(m+M)\,\alpha_{s}\right)-2\,(m+M)\,\alpha_{p}\right)+128\alpha_{p}^{3}\,e\,m\,M\,\alpha_{s}+128\,\alpha_{p}^{4}\,(m+M)\bigg{]}$
(56) $\displaystyle\vspace{0.15cm}c_{9}^{(10)}\ =\
\alpha_{s}\,\bigg{[}\,\alpha_{s}^{3}\,\big{(}\,\alpha_{p}\,\alpha_{s}^{2}\,[\alpha_{s}\left(2e^{2}mM-23\beta
m-21\beta M\right)+4\beta
emM\,]+2M\alpha_{p}^{2}\left(\alpha_{s}\left(\beta+e^{2}m\right)+192\beta
em\right)$
$\displaystyle+\beta\alpha_{s}^{4}\left(28emM+13(m+M)\alpha_{s}\right)\,\big{)}+8\alpha_{p}^{3}\,\alpha_{s}\,[\alpha_{s}\left(5e^{2}mM+58\beta(m+M)\right)+40\beta
emM\,]+160\alpha_{p}^{4}\,\beta\,M\,\bigg{]}$
$\displaystyle\vspace{0.15cm}c_{8}^{(10)}\ =\
\beta\bigg{[}\alpha_{s}^{3}\big{(}2\alpha_{p}\alpha_{s}^{2}[\,\alpha_{s}\left(13e^{2}mM-55\beta
m-42\beta M\right)+212\beta
emM\,]+4\alpha_{p}^{2}\,(\alpha_{s}\left(35e^{2}mM+156\beta m+161\beta
M\right)$ $\displaystyle\ +\
336\,\beta\,e\,m\,M\,)+\alpha_{s}^{4}\left(\alpha_{s}\left(2\,e^{2}\,m\,M+73\,\beta\,m+75\,\beta\,M\right)+172\,\beta\,e\,m\,M\,\right)\big{)}$
$\displaystyle\ +\
16\,\alpha_{p}^{3}\,\alpha_{s}\,[\alpha_{s}\left(9e^{2}mM+26\beta m+64\beta
M\right)+8\,\beta\,e\,m\,M\,]\ +\ 64\,\alpha_{p}^{4}\,\beta\,M\,\bigg{]}$
$\displaystyle\vspace{0.15cm}c_{7}^{(10)}\ =\
\beta^{2}\,\alpha_{s}\,\bigg{[}\,32\,\alpha_{p}^{3}\,M\,\left(26\beta+5e^{2}m\right)+\alpha_{s}\bigg{(}8\alpha_{p}\alpha_{s}^{2}\left(\alpha_{s}\left(33\,e^{2}\,m\,M+11\beta
m+29\beta M\right)+236\beta\,e\,m\,M\,\right)$ $\displaystyle\ +\
8\,\alpha_{p}^{2}\,\big{[}\alpha_{s}\,\left(79\,e^{2}\,m\,M+138\,\beta\,m+256\,\beta\,M\right)+168\,\beta\,e\,m\,M\,\big{]}$
$\displaystyle\ +\
\alpha_{s}^{4}\left(\alpha_{s}\left(26\,e^{2}\,m\,M+231\,\beta\,m+257\,\beta\,M\right)+732\,\beta\,e\,m\,M\,\right)\bigg{)}\bigg{]}$
$\displaystyle\vspace{0.15cm}c_{6}^{(10)}\ =\
2\,\beta^{3}\,\bigg{[}32\,\alpha_{p}^{3}\,M\,(4\,\beta+e^{2}\,m\,)+\alpha_{s}\bigg{(}4\alpha_{p}\alpha_{s}^{2}\left(\alpha_{s}[139\,e^{2}\,m\,M+70\beta\,m+193\,\beta\,M]+376\,\beta\,e\,m\,M\right)$
$\displaystyle\ +\
8\,\alpha_{p}^{2}\,\left(\alpha_{s}\,\left(67\,e^{2}\,m\,M+30\,\beta\,m+166\,\beta\,M\,\right)+24\,\beta\,e\,m\,M\right)$
$\displaystyle\ +\
\alpha_{s}^{4}\,\left(\alpha_{s}\,\left(93\,e^{2}\,m\,M+265\,\beta\,m+338\,\beta\,M\right)+1012\,\beta\,e\,m\,M\,\right)\bigg{)}\bigg{]}$
$\displaystyle\vspace{0.15cm}c_{5}^{(10)}\ =\
2\,\beta^{4}\,\alpha_{s}\,\bigg{[}16M\alpha_{p}^{2}\left(53\beta+26e^{2}m\right)+4\alpha_{p}\alpha_{s}\big{(}\alpha_{s}[283\,e^{2}\,m\,M+64\beta
m+388\,\beta\,M\,]+256\,\beta\,e\,m\,M\,\big{)}$ $\displaystyle\ +\
\alpha_{s}^{3}\,[\,\alpha_{s}\,\left(363\,e^{2}\,m\,M+412\,\beta\,m+707\,\beta\,M\,\right)+1648\,\beta\,e\,m\,M\,]\,\bigg{]}$
$\displaystyle\vspace{0.15cm}c_{4}^{(10)}\ =\
4\,\beta^{5}\,\bigg{[}16\,M\,\alpha_{p}^{2}\left(7\beta+4e^{2}m\right)+4\alpha_{p}\alpha_{s}\,\big{(}\,\alpha_{s}\left(153e^{2}m\,M+8\,\beta\,(m+25\,M)\,\right)+32\,\beta\,e\,m\,M\,\big{)}$
$\displaystyle\ +\
\alpha_{s}^{3}\,\big{(}\,\alpha_{s}\,\left(403\,e^{2}\,m\,M+188\,\beta\,m+549\,\beta\,M\right)+752\,\beta\,e\,m\,M\,\big{)}\,\bigg{]}$
$\displaystyle\vspace{0.15cm}c_{3}^{(10)}\ =\
8\beta^{6}\alpha_{s}\bigg{[}\,4\,M\,\alpha_{p}\,\left(54\beta+43e^{2}m\right)+\alpha_{s}\,\big{(}\,\alpha_{s}\left(265\,e^{2}\,m\,M+44\beta
m+298\beta M\right)+176\beta\,e\,m\,M\big{)}\,\bigg{]}$
$\displaystyle\vspace{0.15cm}c_{2}^{(10)}\ =\
16\,\beta^{7}\,\bigg{[}4\,M\,\alpha_{p}\,\left(6\,\beta+5\,e^{2}\,m\right)+\alpha_{s}\big{(}\alpha_{s}\left(103\,e^{2}\,m\,M+4\beta
m+106\,\beta\,M\right)+16\,\beta\,e\,m\,M\,\big{)}\,\bigg{]}$
$\displaystyle\vspace{0.15cm}c_{1}^{(10)}\ =\
704\,\beta^{8}\,M\,\alpha_{s}\,\left(\,\beta\,+\,e^{2}\,m\,\right)$
$\displaystyle\vspace{0.15cm}c_{0}^{(10)}\ =\
128\,\beta^{9}\,M\,\left(\,\beta\,+\,e^{2}\,m\,\right)$

where $\alpha_{s}\ \equiv\ \alpha_{1}\ +\ \alpha_{2}\quad;\qquad\alpha_{p}\
\equiv\ \alpha_{1}\,\alpha_{2}\ $.

As for the denominator $P_{8}(Z)=\sum_{i=0}^{8}c_{i}^{(8)}\,Z^{i}$, the
coefficients take the form

$\displaystyle c_{8}^{(8)}\ =\
2\,m\,M\,\alpha_{s}^{2}\,\left(\,64\,\alpha_{p}^{3}\ +\
\alpha_{s}^{6}\,\right)$ (57) $\displaystyle c_{7}^{(8)}\ =\
2\,\beta\,m\,M\,\alpha_{s}\,\left(80\,\alpha_{p}^{3}\ +\
\alpha_{p}\,\alpha_{s}^{4}\ +\ 192\,\alpha_{p}^{2}\,\alpha_{s}^{2}\ +\
13\,\alpha_{s}^{6}\,\right)$ $\displaystyle c_{6}^{(8)}\ =\
2\,\beta^{2}\,m\,M\,\left(\,32\,\alpha_{p}^{3}\ +\
202\,\alpha_{p}\,\alpha_{s}^{4}\ +\ 432\,\alpha_{p}^{2}\,\alpha_{s}^{2}\ +\
73\,\alpha_{s}^{6}\,\right)$ $\displaystyle c_{5}^{(8)}\ =\
2\,\beta^{3}\,m\,M\,\alpha_{s}\,\left(336\,\alpha_{p}^{2}\ +\
664\,\alpha_{p}\,\alpha_{s}^{2}\ +\ 295\,\alpha_{s}^{4}\,\right)$
$\displaystyle c_{4}^{(8)}\ =\ 4\,\beta^{4}\,m\,M\,\left(\,48\,\alpha_{p}^{2}\
+\ 424\,\alpha_{p}\,\alpha_{s}^{2}\ +\ 361\,\alpha_{s}^{4}\,\right)$
$\displaystyle c_{3}^{(8)}\ =\
16\,\beta^{5}\,m\,M\,\alpha_{s}\,\left(\,64\,\alpha_{p}\ +\
127\,\alpha_{s}^{2}\,\right)$ $\displaystyle c_{2}^{(8)}\ =\
32\,\beta^{6}\,m\,M\,\left(8\,\alpha_{p}\ +\ 51\,\alpha_{s}^{2}\,\right)$
$\displaystyle c_{1}^{(8)}\ =\ 704\,\beta^{7}\,m\,M\,\alpha_{s}$
$\displaystyle c_{0}^{(8)}\ =\ 128\,\beta^{8}\,m\,M\ .$

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This part is about the notion of almost-fuchsian submanifold. First introduced
for surface group representations in $\PSL(2,\C)$ by Uhlenbeck Uhl83, almost-
fuchsian discs and hypersurfaces have a rich literature
Eps86,KS93,Sep16,San17,Jia21,ES22,BS23,HLS23.

Throughout this section $\X$ will be a Hadamard space with sectional curvature
less or equal than $-1$. Note that we do not require its curvature to admit a
lower bound. We consider an immersion $f:Y\rightarrow\X$, where $Y$ is
connected. We assume $f$ to be complete, i.e. the induced metric on $Y$ is
complete.

We will prove the following theorem. Assume $f:Y\rightarrow\X$ is minimal and
satisfies:

$\sup_{y\in Y}|\II_{f}|<1$ (1)

Then $Y$ is contractible. Denote by $NY$ the normal bundle to $Y$. The
exponential map $\exp:NY\rightarrow\X$ is a diffeomorphism. Also, $f$ extends
at infinity to an embedding from a sphere $S\rightarrow\partial_{\infty}\X$
The proof is divided in several steps. First, we show that the induced metric
by $\exp$ is nondegenerate on $NY$. This will imply that $\exp$ is a covering
map to $\X$, and so it is a diffeomorphism. Finally, we will see that $f$ is a
quasi-isometric embedding from $Y$ to $\X$, so it extends at infinity to an
embedding $S\rightarrow\partial_{\infty}\X$. First note that because of the
condition on $f$, the induced metric has negatively pinched curvature on $Y$,
and it makes sense to talk about its boundary $\partial_{\infty}Y$, which is
constructed in the works of Gromov [Gro78], Eberlein–O’NEill [EO73] and
Ballmann–Gromov–Schroeder [BGS85]. Let $f:Y\rightarrow\X$ be an almost-
fuchsian immersion. Let $\eps>0$ such that

$\sup_{y\in Y}|\II_{f}|\leq 1-\eps\,.$ (2)

Endow $Y$ with the induced metric by $f$. Then its sectional curvature is less
than $-\eps(2-\eps)$. If the sectional curvature of $\X$ is lower bounded by
$-C$, then the sectional curvature of $f^{\ast}g_{\X}$ is bigger than
$-C-2(1-\eps)^{2}$. When $Y$ is a surface, the minimality condition ensures
that we have more precise bounds: The sectional curvature of an almost-
fuchsian surface is then less than $-1$ and bigger than $-C-(1-\eps)^{2}$.
This is a consequence of the Gauss condition: For $u,v$ an orthonormal basis
of a tangent plane to $Y$, By the Gauss’ equation, the curvature of
$f^{\ast}g_{\X}$ is then

$R_{f^{\ast}g_{\X}}(u,v,v,u)=R_{g_{\X}}(u,v,v,u)+g_{\X}(\II_{f}(u,u),\II_{f}(v,v))-g_{\X}(\II_{f}(u,v),\II_{f}(u,v))$
(3)

As $R_{g_{\X}}(u,v,v,u)\leq-1$ and $|\II_{f}(u,u)|\leq 1-\eps$, we directly
get

$R_{f^{\ast}g_{\X}}(u,v,v,u)\leq-1+(1-\eps)^{2}=-\eps(2-\eps)\,.$ (4)

as desired. Moreover, if $R_{g_{\X}}(u,v,v,u)\geq-C$, we directly get that

$R_{f^{\ast}g_{\X}}(u,v,v,u)\geq-C-2(1-\eps)^{2}\,,$ (5)

as claimed. Let $f:Y\rightarrow\X$ be an almost-fuchsian immersion. Denote by
$NY$ the normal bundle to $Y$, and note $G$ the induced metric by the
exponential map on $NY$. For $t>0$, denote by $N_{t}Y$ the tangent bundle to
the distance $t$ hypersurface to $Y$. Let $(x,tv)\in NY$ with $v$ a unit
normal vector. Let $B_{t}$ be the parallel transport map from
$T_{x}\X\cap(v)^{\perp}$ to $T_{\exp{tv}}(N_{t}Y)$. Let $w\in T_{v}(UY)$ and
let $f$ be the function:

$f(t)=|B_{t}w|^{2}$ (6)

If $\X$ is the hyperbolic space, $f$ satisfies the following controls:

$(\sqrt{f})^{\prime\prime}=\sqrt{f}$ (7)

If $\X$ has sectional curvature strictly less than $-1$, then

$(\sqrt{f})^{\prime\prime}>\sqrt{f}$ (8)

Also, in both cases

$(\ln f)^{\prime\prime}+\frac{(\ln f)^{\prime 2}}{2}\geq 2$ (9)

with the same equality and inequality conditions. With the introduced
notations, fix $\gamma:I\rightarrow UY$ a smooth path such that $\gamma(0)=v$
and $\gamma^{\prime}(0)=w$. The family $c_{s}(t)$ is a smooth variation of
geodesics, hence $J(t)=\partial_{s}c_{s}(t)|_{s=0}$ is a Jacobi field along
$c_{0}$. Evaluate the characteristic equation of Jacobi fields against $J$:

$g_{\X}(\ddot{J},J)+R_{\X}(J,\partial_{t}c,\partial_{t}c,J)=0$ (10)

In the second term, we recognize the sectional curvature of the tangent plane
spanned by $J$ and $\partial_{t}c$. Choose a local orthonormal chart of
$T_{x}\X$ such that $B_{0}=I_{k}\oplus O_{n-k-1}$ and $\dot{B}_{0}=B(v)\oplus
I_{n-k-1}$, where $B(v)$ is the shape operator of $f$ at $v$. Then equation 10
rewrites as

${}^{\perp}w\ddot{B}_{t}B_{t}w+R_{\X}(J,\partial_{t}c,\partial_{t}c,J)=0$ (11)

and as the sectional curvature is less than $-1$, we replace the second term
with the control

$R_{\X}(J,\partial_{t}c,\partial_{t}c,J)\leq-^{\perp}w^{\perp}B_{t}B_{t}w$
(12)

to get the following

${}^{\perp}w^{\perp}(\ddot{B}_{t}-B_{t})B_{t}w\geq 0$ (13)

Now remark that

$\left\\{\begin{array}[]{ll}f^{\prime}(t)&=2^{\perp}w^{\perp}B_{t}\dot{B}_{t}w\\\
f^{\prime\prime}(t)&=2^{\perp}w^{\perp}B_{t}\ddot{B}_{t}w+2^{\perp}w^{\perp}\dot{B}_{t}\dot{B}_{t}w\end{array}\right.$
(14)

So the characteristic equation rewrites as

$f^{\prime\prime}(t)\geq 2f(t)+\frac{f^{\prime}(t)^{2}}{2f(t)}$ (15)

with equality when $\X$ is the hyperbolic space, and strict inequality if $\X$
has sectional curvature strictly less than $-1$. As a direct consequence,

$(\sqrt{f})^{\prime\prime}\geq\sqrt{f}$ (16)

with the same equality and inequality conditions. Finally, introduce
$g=\ln(f)$. As long as it is well defined, we deduce from 15 that

$g^{\prime\prime}+\frac{(g^{\prime})^{2}}{2}\geq 2$ (17)

When $\X$ has varying curvature less or equal than $-1$, up to rescaling the
metric we can assume its sectional curvature to be strictly less than $-1$.
[Proof of theorem 1] With the notations introduced, Fix $w\in T_{v}(UY)$ and
consider the function $f$ defined in proposition 1. Denote $w_{1}$ the
projection of $w$ on $TY$, and assume it is nonzero. The initial conditions,
by construction of $f$,

$\left\\{\begin{array}[]{ll}f(0)&=\alpha^{2}=|w_{1}|^{2}>0\\\
f^{\prime}(0)&=2\alpha\beta=2^{\perp}w_{1}B(v)w_{1}<2\alpha\end{array}\right.$
(18)

From proposition 1, we deduce that

$\left\\{\begin{array}[]{lll}f(t)&\geq(\alpha\cosh(t)+\beta\sinh(t))^{2}\\\
f^{\prime}(t)&\geq
2(\alpha\cosh(t)+\beta\sinh(t))(\alpha\sinh(t)+\beta\cosh(t))\\\
g^{\prime}(t)&=\frac{f^{\prime}(t)}{f(t)}\geq
2\frac{\alpha\sinh(t)+\beta\cosh(t)}{\alpha\cosh(t)+\beta\sinh(t)}\end{array}\right.$
(19)

with equality when $\X$ is hyperbolic, and strict inequality when the
curvature of $\X$ is stricly less than $-1$. A first consequence is that $f$
never vanishes. Hence $(B_{t})$ has trivial kernel, and the metric induced by
$\exp$ on $NY$ is nondegenerate. So $\exp$ is a local diffeomorphims from $NY$
to $\X$. Moreover, because the immersion $Y\rightarrow\X$ is complete, it is
proper. Hence for any point of $x\in\X$ there is at least one point in $Y$
which minimizes the distance to $x$. This implies that the map $\exp$ is
surjective. As a proper surjective local diffeomorphism, $\exp$ satisfies the
path lifting property, so it is a covering map. But $Y$ is connected and $\X$
is contractible, so $\exp$ is a global diffeomorphism, $Y$ is embedded and
contractible too. It remains to prove that $f$ is a quasi-isometric embedding.
In order to prove we introduce a metric on $NY$ for which the inverse map of
the exponential map will be Lipschitz, and such that $Y$ sits in $NY$ totally
geodesically with the induced metric by $f$: Consider $(x,tv)\in NY$ and
decompose its tangent space into $T_{(x,tv)}NY=T_{t}(x,\R v)\oplus
T_{x}Y\oplus T_{v}\S^{n-k-1}$. Introduce the metric $h$ on $NY$ which in that
decomposition is:

$h_{(x,tv)}=1\oplus\cosh^{2}(t)f^{\ast}g_{\X,x}\oplus\sinh^{2}(t)g_{\S^{n-k-1}}$
(20)

By construction, $Y$ sits inside $NY$ totally geodesically with the metric
induced by $f$, that we denote $g_{Y}$. The equation on $f$ 19 ensures that:

$\exp^{\ast}g_{\X}\geq
1\oplus(\cosh(t)I_{k}+B(v)\sinh(t))^{2}g_{Y}\oplus\sinh^{2}(t)g_{\S^{n-k-1}}$
(21)

Denote $\delta=1-\sup|\II_{f}|>0$. Then

$\exp^{\ast}g_{\X}\geq\delta^{2}(1\oplus\cosh^{2}(t)g_{Y}\oplus\sinh^{2}(t)g_{\S^{n-k-1}})=\delta^{2}h$
(22)

Hence the inverse map of the exponential map is $\frac{1}{\delta}$-Lipschitz.
As a consequence, $f$ is a quasi-isometric embedding, as it satisfies:

$d_{\X}(f(x_{1}),f(x_{2}))\leq
d_{Y}(x_{1},x_{2})\leq\frac{1}{\delta}d_{\X}(f(x_{1}),f(x_{2}))$ (23)

Combined with the fact that $Y$ has negatively pinched sectional curvature
proven in lemma 1, it ensures that $f$ extends at infinity to an embedding
from a sphere $\S^{k-1}\rightarrow\partial_{\infty}\X$. As a corollary of the
computations we made, we have some explicit estimates of the geometry of $NY$,
endowed with the induced metric by $\exp$. Let $f:Y\rightarrow\X$ be a
complete almost-fuchsian immersion, with $\X$ a Hadamard space. Denote by
$N_{t}Y$ the distance $t$ hypersurface to $Y$. At a point $\exp_{x}(tv)\in
N_{t}Y$, consider $\lambda_{1}\leq\ldots\leq\lambda_{k}$ the eigenvalues of
$B(v)$ the shape operator of $Y$ evaluated at $(x,v)$. Then the second
fundamental form $\II_{t}$ of $N_{t}Y$ has eigenvalues
$\lambda_{1}^{t}\leq\ldots\leq\lambda_{n-1}^{t}$ which satisfy ∀i≤k,
∑_j=1^iλ_j^t &≥∑_j=1^i λj+tanh(t)1+λjtanh(t)
∀i¿ k, ∑_j=1^iλ_j^t ≥∑_j=1^k λj+tanh(t)1+λjtanh(t)+(i-k)1tanh(t) In
particular, $N_{t}Y$ is $k$-convex in the sense of Sha [Sha86]. This is a
consequence of the control on $\frac{f^{\prime}}{f}$ gotten in 19. The trace
of $\II_{t}$ on a $k$-plane $P$ is the sum of
$\frac{1}{2}\frac{f^{\prime}(t)}{f(t)}$ for $w$ spanning a basis of the plane
$P$. The almost-fuchsian condition ensures that the sum of the $\lambda_{i}$
is 0, and that they live in a compact set of $(-1,1)$, which ensures the
$k$-convexity of $\II_{t}$.