††thanks: ORCiD: 0000-0002-6286-0722††thanks: ORCiD: 0000-0002-4509-4309

# Study on the Accuracy of the Envelope Theory

Lorenzo Cimino<EMAIL_ADDRESS>Cyrille Chevalier
<EMAIL_ADDRESS>Ethan Carlier<EMAIL_ADDRESS>Joachim Viseur<EMAIL_ADDRESS>Service de Physique
Nucléaire et Subnucléaire, Université de Mons, UMONS Research Institute for
Complex Systems, Place du Parc 20, 7000 Mons, Belgium

###### Abstract

Abstract The envelope theory is an easy-to-use approximation method to obtain
eigensolutions for some quantum many-body systems. Even if the solutions are
reliable and an improvement procedure exists, the method can lack of accuracy
for some systems. In a previous work, two hypotheses were proposed to explain
the low precision: the presence of a divergence in the potential and a mix of
variational characters. In the present work, different systems are studied to
test these hypotheses. Theses tests show that the presence of a divergence
causes indeed less accurate results, while the mix of variational characters
reduces the impact of the improvement procedure.

Envelope theory; Many-body quantum systems; Approximation methods

##  I Introduction

The resolution of many-body quantum systems equation is always a challenging
task. Many approximate computational methods exist, each with their strengths
and weaknesses. Among them, the envelope theory (ET) [3, 4, 5], also known as
the auxiliary field method [6, 7], is an easy to use method with the following
advantages: (i) it can treat quite general Hamiltonians with non-standard
kinematics in $D$ dimensions; (ii) its computation cost is independent of the
number of particles; (iii) in favourable cases, the approximate eigenvalues
can be lower or upper bounds.

The initial idea of the method is to replace the Hamiltonian $H$ under study
by an auxiliary Hamiltonian $\tilde{H}$ which is solvable [7]. In practice,
$\tilde{H}$ is chosen as a many-body harmonic oscillator $H_{\text{ho}}$,
which is exactly solvable for arbitrary dimensions [8, 9]. The auxiliary
Hamiltonian, as its name implies, depends on some auxiliary parameters which
are fixed by an extremisation procedure. It is worth noting that,
historically, the ET has been developed as a “geometrical theory” which is
completely equivalent to the introduction of the above auxiliary Hamiltonian,
and will be explained in more details below. Beyond the auxiliary Hamiltonian,
an improvement procedure has been developed based on a modification of the
harmonic oscillator quantum numbers associated with the ET [10].

The ET has been used to obtain physical results about hadronic systems as in
[11]. The method is especially useful when the number of particles can be
arbitrary large as in the large-$N_{c}$ formulation of QCD [12, 13]. The
method has also allowed the study of a possible quasi Kepler’s third law for
quantum many-body systems [14]. Finally, the method can also be simply used to
test accurate numerical calculations as in [15].

Even if the method is reliable, its accuracy is not predictable. Previous
tests [16, 17, 18] showed that the ET leads to low relative errors for some
favourable systems such as linear or Gaussian potentials. On the other hand,
for systems with a Coulomb potential the ET has a lower accuracy, but the use
of the improvement procedure allows to recover pretty good results. However,
when studying atomic spectra, both the ET and the improvement failed to
reproduce the binding energies.

The aim of this work is to find qualitative features to predict the accuracy
of the ET. Such features would improve its attractiveness. In Sec. II, the
basics of the ET are presented, putting the emphasis on practical angles.
Similarly, Sec. III shortly presents the improved ET, emphasising again
practical requirements. Sec. IV is entirely dedicated to accuracy tests and
their interpretations. The three first subsections investigate the impact of
divergences on the ET, while the three last subsections focuses on the role of
variational character. Finally, Sec. V sums up the initial issue while
concluding about the above-mentioned qualitative features.

##  II Envelope theory

###  II.1 Compact equations

The ET is an approximation method able to determine the spectrum of generic
$N$-body Hamiltonians. Let us consider a system of $N$ identical particles
governed by the following Hamiltonian,

$H=\sum_{i=1}^{N}T(p_{i})+\sum_{i<j=2}^{N}V(r_{ij}),$ (1)

where $p_{i}=\left\lvert\bm{p_{i}}\right\rvert$ and
$r_{ij}=\left\lvert\bm{r}_{i}-\bm{r}_{j}\right\rvert$. The functions $T(x)$
and $V(x)$ are respectively interpreted as a kinetic energy and a two-body
potential. These are general functions that must nevertheless satisfy a few
constraints [19] which are not much restrictive and in agreement with the
physical interpretation of $T$ and $V$. The ET ensures that a good
approximation for the spectrum of this Hamiltonian is given by the resolution
of the following set of three equations [20],

$\displaystyle E=N\,T(p_{0})+C^{2}_{N}\,V(\rho_{0}),$ (2a) $\displaystyle
N\,T^{\prime}(p_{0})\,p_{0}=C^{2}_{N}\,V^{\prime}(\rho_{0})\,\rho_{0},$ (2b)
$\displaystyle Q(N)=\sqrt{C^{2}_{N}}\,p_{0}\,\rho_{0},$ (2c)

where $f^{\prime}$ denotes to the derivative of $f$ with respect to its
argument and where $C^{2}_{N}=N(N-1)/2$ is the number of pairs of particles.
The equations are written in natural units ($c=\hbar=1$). In this set,
commonly called compact equations of the ET, the equation (2a) gives an
expression of the approximated spectrum $E$ in terms of two unknown parameters
$p_{0}$ and $\rho_{0}$. The two remaining equations allows to determine
$p_{0}$ and $\rho_{0}$ in terms of a global quantum number $Q(N)$. The latter
is related to radial and orbital quantum numbers $n_{\alpha}$ and
$l_{\alpha}$, related to the internal Jacobi variables, as follows

$Q(N)=\sum\limits_{\alpha=1}^{N-1}\left(2n_{\alpha}+l_{\alpha}+\frac{D}{2}\right).$
(3)

Let us notice that the dimension of the system, $D$, is arbitrary in the ET
and that $Q(N)$ includes its only occurrence in the equations. To be complete,
in the case of one dimensional systems, the expression of $Q(N)$ must be
adapted due to the absence of angular momentum [17, 21],

$Q(N)=\sum\limits_{\alpha=1}^{N-1}\left(n_{\alpha}+\frac{1}{2}\right).$ (4)

Finally, we can mention that the $p_{0}$ and $\rho_{0}$ parameters obtained
during the resolution of (2) approximate instructive mean values in the
system,

$\displaystyle p_{0}^{2}=\left\langle p_{i}^{2}\right\rangle,$
$\displaystyle\rho_{0}^{2}=\left\langle r_{ij}^{2}\right\rangle.$ (5)

Because all the particles are identical, the $i$ and $j$ subscripts can be
chosen indiscriminately in $\\{1,...,N\\}$. Let us mention that in previous
papers, the parameter
$r_{0}^{2}=N^{2}\left\langle(\bm{r}_{i}-\bm{R})^{2}\right\rangle$ was used
instead of $\rho_{0}$ allowing the treatment of one-body potential. One can
show that for identical particles $r_{0}^{2}=C^{2}_{N}\rho_{0}^{2}$. Since
only two-body potential will be considered here, it is preferable to use
$\rho_{0}$.

###  II.2 Quick review of the theory behind the method

Even though the aim of this article is not to demonstrate the above equations
(the interested reader can look in [7] for the full details or [22] for a
summary), let us here recap the main concepts about the construction of the
method. The ET suggests to approximate a $N$-body Hamiltonian satisfying the
structure (1) by a set of $N$-body harmonic oscillator Hamiltonians, called
auxiliary Hamiltonians. Each of them are chosen so that they approximate one
of the energy levels of the genuine Hamiltonian. As the $N$-body harmonic
oscillator spectrum can be analytically computed, the set of auxiliary
Hamiltonians provides an analytical approximation for the genuine spectrum.
The appearance in the compact equations of $Q(N)$, the quantum number of the
spectrum of the $N$-body harmonic oscillator, is a vestige of these auxiliary
Hamiltonians. Let us also mention that the mean values in (5) are taken with
eigenstates of these auxiliary harmonic oscillators.

Eventually, the auxiliary set is obtained by an extremisation procedure on the
energy (a justification of this procedure can be found in [7]) which can be
shown equivalent to the three compact equations of the ET (2). It can also be
shown that each part of the auxiliary Hamiltonians obtained by this
extremisation process are tangent to their respective parts of the genuine
Hamiltonian. This property gave historically its name to the method.

###  II.3 Variational Character

For some favourable Hamiltonians, the approximated energy given by the ET can
be interpreted as an upper or lower bound for the original spectrum [20, 7,
22]. This variational character is provided by the tangentness of the
auxiliary Hamiltonians together with the comparison theorem [23]. Let us
define the two following functions,

$\displaystyle b_{T}(x^{2})=T(x),$ (6) $\displaystyle b_{V}(x^{2})=V(x).$ (7)

If both the $b_{T}$ and $b_{V}$ functions are concave (resp. convex) the
approximated spectrum given by the ET is an upper (resp. lower) bound. If the
second derivative of one of these functions vanishes the variational character
is governed by the other. If both the second derivative of $b_{T}$ and $b_{V}$
function vanish, the original Hamiltonian was an harmonic oscillator and the
approximation given by the ET is actually the exact spectrum. Nevertheless, if
one of the functions does not have a universal concavity on its domain or if
the concavity of two functions are not the same, no variational character can
be defined. Examples of computations of functions $b_{V}$ and $b_{T}$ will be
given in the next section.

Before concluding the section, it is worth noting that generalisations of the
ET exist. The approximation method can deal with Hamiltonian including one-
body potentials and some kind of $K$-body forces [22]. Furthermore, the method
has also been recently generalised to systems of $N_{a}$ identical particles
of type $a$ and $N_{b}$ identical particles of type $b$ [18, 24]. Both
generalisations lead to a new set of compact equations (2) but the rest of the
discussion remains relevant.

##  III Improvement of the envelope theory

Despite the approximation given by the ET is reliable, it can lack of accuracy
for some Hamiltonians. To sort it out, a parameter $\phi$ can be added in the
global quantum number $Q(N)$,

$Q_{\phi}(N)=\sum_{\alpha=1}^{N-1}\left(\phi
n_{\alpha}+l_{\alpha}+\frac{D+\phi-2}{2}\right).$ (8)

This new feature is inspired by [25] where two-body systems are considered. It
is worth noting that $Q(N)$ is recovered by setting $\phi=2$. When a part of
the spectrum is accurately known (e.g. thanks to a previous very precise
calculation), $\phi$ can be fixed by fitting the ET approximation on this
level. However $\phi$ can also be calculated from scratch thanks to the
coupling of the ET with a generalisation of the dominantly orbital state
method to $N$-body systems. This coupling leads to the following analytical
expression for $\phi$,

$\displaystyle\phi=$
$\displaystyle\frac{\lambda}{N\tilde{p}_{0}T^{\prime}(\tilde{p}_{0})}\sqrt{\frac{k}{C_{N}^{2}\mu}},$
(9) $\displaystyle\text{with
}\mu=\frac{\tilde{p}_{0}}{NT^{\prime}(\tilde{p}_{0})},$ (10)
$\displaystyle\text{with
}k=\frac{2N\tilde{p}_{0}}{\tilde{\rho}_{0}^{2}}\,T^{\prime}(\tilde{p}_{0})+\frac{N\tilde{p}_{0}^{2}}{\tilde{\rho}_{0}^{2}}\,T^{\prime\prime}(\tilde{p}_{0})+C_{N}^{2}\,V^{\prime\prime}(\tilde{\rho}_{0}^{2}),$
(11) $\displaystyle\text{and where
}\begin{cases}N\,T^{\prime}(\tilde{p}_{0})\,\tilde{p}_{0}=C^{2}_{N}\,V^{\prime}(\tilde{\rho}_{0})\,\tilde{\rho}_{0},\\\
\sqrt{C^{2}_{N}}\,\tilde{p}_{0}\,\tilde{\rho}_{0}=\sum_{\alpha=1}^{N-1}\left(l_{\alpha}+\frac{D-2}{2}\right).\end{cases}$
(12)

The expression given here is equivalent to the one in [10] but is shorter due
to an unnoticed simplification. It is worth noting that the two equations (12)
are actually the two last compact equations of the ET (equations (2b) and
(2c)) with $Q(N)$ replaced by another combination of the internal quantum
numbers which only includes the orbital ones. When $\phi$ is computed, one can
start a second resolution of the compact equation (2) but with $Q(N)$ replaced
by $Q_{\phi}(N)$. This procedure has been tested in [16] for a few physical
potentials concluding that, indeed, the addition of $\phi$ improves the
accuracy of the ET.

Regarding generalisations, the improvement has also been developed for systems
of $N_{a}$ identical particles plus one different [26] and has been written
for Hamiltonians including one-body potentials (only in the case of $N$
identical particles). Other generalisations seem possible but have still to be
written. It is worth noting that, as the name of the method suggests, the
dominantly orbital state method requires $D>1$. Consequently, the improvement
of the ET presented here can not deal with one-dimensional systems. Before
moving on, it is also worth noting that from the modification of $Q(N)$, the
variational character, if it exists, can no longer be guaranteed. In the
following, the designation “classical ET” will refer to the ET without the
additional parameter $\phi$, in contrast to “improved ET” which refers to the
version that includes it.

##  IV Accuracy tests

We saw in the previous sections that the ET is an easy-to-use approximation
method whose accuracy is nevertheless difficult to predict. We know that the
improved version of the method should make this accuracy better but it remains
difficult to quantify it. Especially some recent tests about a generalisation
of the method for systems of different particles revealed a low accuracy of
the ET in the reproduction of atomic spectra [18, 26]. In the corresponding
articles, two hypotheses were formulated to explain such a poor accuracy:

  * •

the divergence of the coulomb interaction is difficult to reproduce with
harmonic oscillators,

  * •

the mix between attractive and repulsive potentials prevents having
variational character.

The rest of this article is dedicated to test if these two situations have
indeed an impact on the accuracy of the ET. For simplicity, we focus on
systems of three identical particles with non-relativistic kinematics
$T(p)=p^{2}/2m$. In addition, only bosonic ground states (BGS) at $D=3$
dimensions are considered. However, in the following, formula are given with
an arbitrary number of particles $N$, mass $m$ and global quantum number
$Q_{\phi}(N)$ to stay general. But, in the graphs, $N$ is fixed to $3$, $m$ is
fixed to $1$ and the modified global quantum number for BGS reduces to

$Q_{\phi}^{BGS}(N=3)=\phi+1,$ (13)

where it is worth recalling that $\phi=2$ leads to the original global quantum
number $Q(N)$. Since the second derivative of $T(p)$ vanishes, the variational
character is solely given by the potential. Before to dive into the tests, let
us mention that the ET approximations will be compared to the results from a
very accurate numerical method based on an oscillator basis expansion [27,
28], whose numerical cost is nevertheless considerably higher than the one of
the ET. Because this method is several order of magnitude more accurate than
the ET, its results are referred below as “exact results”.

###  IV.1 Negative power potential

To test the impact of the divergence we consider a negative power-law
potential $V(r)=-G\,r^{\beta}$, with $\beta<0$. Indeed, the bigger
$\left\lvert\beta\right\rvert$ is, the more divergent the potential is. The
approximation given by the ET for this potential have already been computed in
[26] for a more general system (including a power-law kinematics). For non-
relativistic kinematics, it reduces to

$\displaystyle\rho_{0}=\left(\frac{N\,Q_{\phi}(N)^{2}}{m|\beta|G(C^{2}_{N})^{2}}\right)^{\frac{1}{2+\beta}},$
(14a) $\displaystyle
E=-(2+\beta)\left[\left(\frac{G}{2}\right)^{2}(C^{2}_{N})^{2-\beta}\left(\frac{N\,Q_{\phi}(N)^{2}}{2m|\beta|}\right)^{\beta}\right]^{\frac{1}{2+\beta}},$
(14b)

with $\phi=\sqrt{2+\beta}$. It has been shown in the same article that this
result is a lower bound. For $\beta=-1$, that is the Coulomb potential, the
expected result from [16] is recovered.

Comparisons with the exact energies are presented in Fig. 1 for $G=1$ and for
some values of $\beta$. It can be seen that for small
$\left\lvert\beta\right\rvert$, the ET is quite accurate. Nevertheless, the
relative error rises as $\left\lvert\beta\right\rvert$ increases. On the other
hand, with the improved ET, the behaviour with $\beta$ is well reproduced. For
$\beta$ close to $-2$, the accuracy of improved ET result seems to decrease.
However, we can note that, in such a range of $\beta$, the results from the
oscillator basis expansion should be treated cautiously because convergence
was not achieved.

Figure 1: Energy $E$ of a three-body system with $m=1$ and a negative power
potential $V(r)=-r^{-\beta}$ as a function of $\left\lvert\beta\right\rvert$.
Exact results (blue), ET results (orange) and improved ET results (green).

###  IV.2 Truncated Coulomb potential

Another way to test the impact of the divergence is to turn it off using a
truncation. The Coulomb potential can be modified by introducing a bias
distance $d$ such that it is truncated at the origin: $V(r)=-c/(r+d)$, with
$c>0$. An analytical solution of the compact equations (2b) and (2c) exists
but it requires the roots of a cubic equation of the form $t^{3}\pm 3t-2Y=0$.
We will limit ourselves to denote these $F_{\pm}(Y)$, their explicit
expression being written in [7, 29]. After insertion of $\rho_{0}$ and $p_{0}$
in (2a), the following approximated energies are obtained,

$\displaystyle\rho_{0}=d\,y$ (15a) $\displaystyle
E=-\frac{1}{C^{2}_{N}}\frac{c}{d}\frac{y+2}{2(y+1)^{2}},$ (15b)
$\displaystyle\text{with
}y=\frac{\sqrt{A(A+6)}}{3}F_{-}\left(\frac{2A^{3}+18A^{2}+27A}{2(A(A+6))^{3/2}}\right)+\frac{A}{3}\text{
and }A=\frac{N}{(C^{2}_{N})^{2}}\frac{Q_{\phi}(N)^{2}}{m\,c\,d}.$ (15c)

This energy can be shown as an upper bound of the genuine spectrum. Even if
the analytical expression of the spectrum given by the ET is complicated to
write (mainly due to the $F_{\pm}(Y)$ functions) it is worth nothing that it
can simplify in some limits. For instance, in the limit $d>>r$ and $d<<r$,
(15) reduces to

$\displaystyle
E_{d>>r}=-C^{2}_{N}\frac{c}{d}+\frac{3}{2}\left(\frac{N}{C^{2}_{N}}\right)^{1/3}\left(\frac{Q_{\phi}(N)^{2}c^{2}}{m\,d^{4}}\right)^{1/3},$
(16a) $\displaystyle
E_{d<<r}=-\frac{(C^{2}_{N})^{3}}{N}\frac{mc^{2}}{2Q_{\phi}(N)^{2}}+\frac{(C^{2}_{N})^{4}}{N^{2}}\frac{m^{2}c^{3}d}{Q_{\phi}(N)^{4}}.$
(16b)

Since an analytical solution for $\rho_{0}$ and $p_{0}$ exists, one could also
in theory obtain an analytical expression for $\phi$ by using (9) - (12).
However, due to the complexity of the above solution, the value of $\phi$
would be too complex to present here and a numerical evaluation turned out to
be more comfortable and faster.

By varying $d$, the divergence of the potential can be controlled. For $d=0$,
the original Coulomb potential is of course recovered. Accuracy tests are
presented in Fig. 2 for $c=1$. The more $d$ increases, the more accurate the
ET is. Again, for the improved ET, the behaviour is almost exactly reproduced.

Figure 2: Energy $E$ of a three-body system with $m=1$ and a truncated Coulomb
potential $V(r)=-1/(r+d)$ as a function of $d$. Exact results (blue), ET
results (orange) and improved ET results (green).

###  IV.3 Exciton potential

A smoother way to truncate the Coulomb potential is to replace it by
$V(r)=-c/\sqrt{r^{2}+d^{2}}$, with $c>0$. This potential is used to model
electron-hole pairs (also known as excitons) [30]. Equations (2c) and (2b)
yield

$\frac{m\,c}{N}\frac{(C^{2}_{N})^{2}}{Q_{\phi}(N)^{2}}\rho_{0}^{4}=(\rho_{0}^{2}+d^{2})^{3/2}.$
(17)

Since the solutions of this quartic equation are complicated to manipulate,
numerical computations are preferred. It can be shown that the ET provides
upper bounds for this potential. This system has already been studied with the
ET in [21] but only at $D=1$ for two-body systems. As the ET results are
numerically obtained, it is also the case for the improved ET.

Figure 3: Energy $E$ of a three-body exciton system with $m=1$ and
$V(r)=-1/\sqrt{r^{2}+d^{2}}$ as a function of $d$. Exact results (blue), ET
results (orange) and improved ET results (green).

The conclusion for this potential are quite similar to the ones of the
previous example. The relative errors decrease when the divergence is smoothed
(i.e. when $d$ increases). Concerning the improved results, they reproduce
better the $d$ dependence of energy.

###  IV.4 Cubic and linear potential

The second hypothesis proposed to explain the poor accuracy of the ET in the
evaluation of atomic spectra was about its possible variational character. In
atomic systems, the repulsive Coulomb interactions of the electrons tend to
give a lower bound while the attractive Coulomb interactions with the nucleus
tend to give an upper bound. As a result, no prediction can be provided about
the variational character of the ET for atomic systems.

Such a situation can be reproduced by adding together a first potential
leading to an upper bound and a second one leading to a lower bound.
Furthermore, to handle the presence of the variational character, a constant
$C$ is added so that by varying continuously $C$ from $0$ to $1$, one
potential is “turned on” while the other one is “turned off”. Consequently,
going from $C=0$ to $C=1$ amounts to going from a variational character to the
other one.

Let us apply it to a first example. On one hand, the cubic potential
$V(r)=\alpha\,r^{3}$ produces a lower bound because its function
$b_{V}(r)=\alpha\,r^{3/2}$ is convex. On the other hand, the linear potential
$V(r)=\beta\,r$ produces an upper bound since its function
$b_{V}(r)=\beta\,r^{1/2}$ is concave. Therefore, the following potential, that
mixes both previous ones, will present preferentially, depending on the value
of $C$, one variational character or the other one,

$V(r)=\alpha\,C\,r^{3}+\beta\,(C-1)\,r.$ (18)

When $C=0$, $V(r)$ reduces to the linear potential and when $C=1$, $V(r)$
reduces to the cubic potential. To get the approximated spectrum, the compact
equations (2) must be solved once again. It yields the following quintic
equation,

$\frac{N}{m}\frac{Q(N)^{2}}{(C^{2}_{N})^{2}}-3\alpha\,C\rho_{0}^{5}-\beta(1-C)\rho_{0}^{3}=0,$
(19)

which has no easy-to-use analytical solution. Numerical computations will then
be performed for both the ET and its improvement.

Figure 4: Energy $E$ of a three-body system with $m=1$ and a mixed cubic -
linear potential $V(r)=C\,r^{3}+(1-C)r$ as a function of $C$. Exact results
(blue), ET results (orange) and improved ET results (green).

Results from these computations are gathered in Fig. 4 ($\alpha$ and $\beta$
are set to $1$). As expected, when $C=0$ (pure linear potential), the
classical ET provides an upper bound for the energy. On the other extreme,
when $C=1$ (pure cubic potential), it is a lower bound which is provided. For
both cases, the results from the improved ET are noticeably more accurate.

In between these two values of $C$ the situation is different. The errors in
the evaluation of the spectrum produced by the two parts of the potential
counterbalance each other, resulting in a more precise energy. The height of
this effect occurs around $C=0.2$ where the classical ET results go through
the exact one (accidentally).

Furthermore, the improved ET seems far less efficient in between $C=0$ and
$C=1$ than at the extremes. Variational character can be understood as a
safeguard that prevents ET approximations dropping too low (or rising too
high). It seems that, because of the loss of such a variational guidance, the
improved ET has difficulties making its accuracy better. In conclusion, it
seems that in such a situation it is not guarantee that the improved ET is
really more precise than the classical one. This outcome should be slightly
qualified because the dashed curve reproduces better the exact shape and
because our analysis is only interested in the bosonic ground state.

###  IV.5 Cubic and logarithmic potential

To confirm the analysis, a second potential is studied. Considering that a
logarithmic interaction $V(r)=\beta\ln(r)$ also produces an upper bound (the
associated $b_{V}(r)=\beta\ln(\sqrt{r})$ function is concave), the linear
contribution in the previous potential can be replaced by a logarithmic one
while keeping the mix between variational characters,

$V(r)=\alpha\,C\,r^{3}+\beta(1-C)\ln{(r)}.$ (20)

The associated compact equations (2) yield the following quintic equation
which has no trivial solution,

$\frac{N}{m}\frac{Q(N)^{2}}{(C^{2}_{N})^{2}}-3\alpha\,C\rho_{0}^{5}-\beta(1-C)\rho_{0}^{2}=0.$
(21)

Again, numerical calculations will be performed for both the ET and its
improvement.

The results of this second test are presented in Fig. 5 for $\alpha=\beta=1$.
They confirm the previous conclusions. When $C\in\\{0,1\\}$, the classical ET
has the expected variational character and its improvement is really more
accurate. But in between these two values, the upper and lower bounds
counterbalance each other, resulting in a more accurate prediction given by
the classical ET. Concerning the improved version of the method, its results
are again not systematically more accurate than the original ones. In this
example, it is not even clear if the shape of the curve is better reproduced
by the ET or by its improvement.

Figure 5: Energy $E$ of a three-body system with $m=1$ and a mixed Cubic -
logarithmic potential $V(r)=C\,r^{3}+(1-C)\ln{r}$ as a function of $C$. Exact
results (blue), ET results (orange) and improved ET results (green).

###  IV.6 Cubic and Gaussian potential

Finally, a last test about mixes of variational character can be performed.
Just like the linear and logarithmic potentials, the Gaussian potential
$V(r)=-\alpha e^{-r^{2}}$ contributes as an upper bound. The following
combination mixes therefore variational characters as well,

$V(r)=\alpha\,C\,r^{3}-\beta(1-C)\,e^{-r^{2}}$ (22)

The simplification of the compact equations (2) yields to the following
equation whose solution cannot be found analytically,

$\frac{N}{m}\frac{Q(N)^{2}}{(C^{2}_{N})^{2}}-3\alpha\,C\rho_{0}^{5}+2\beta(1-C)e^{-\rho_{0}^{2}}\rho_{0}^{4}=0.$
(23)

Numerical computations will once again be necessary for both the ET and its
improvement. It is worth noting that the constant $\beta$ must be chosen
higher than one to ensure the existence of a bound state.

Figure 6: Energy $E$ of a three-body system with $m=1$ and a mixed Cubic -
Gaussian potential $V(r)=C\,r^{3}-10(1-C)e^{-r^{2}}$ as a function of $C$.
Exact results (blue), ET results (orange) and improved ET results (green).

The obtained energies are summarised in Fig. 6 for $\alpha=1$ and $\beta=10$.
The situation looks here slightly different. If the results from the classical
ET are nearly the same as the ones of the two other tests, the improved ET
gives energies which seem almost always a bit more accurate than the one of
the original method. Both approximation methods cross the exact result curve
around the same value of $C$. It heightens the impression that what occurs
when variationnal character are mixed cannot be predicted.

##  V Concluding remarks

The ET is a reliable and easy to implement approximation method for $N$
identical particle systems [6, 7, 20]. Its accuracy is nevertheless difficult
to predict. For some Hamiltonians, the ET presents a variational character,
guiding through the exact result. Recently, the ET and its improvement have
been generalised for systems of $N_{a}$ identical particles of type $a$ and
one particle of type $b$ [18, 24]. Most tests performed with this
generalisation proved to be conclusive but the study of atomic spectra
revealed an unexpected low accuracy, even after improvement [26]. Two
hypotheses were advanced to explain this phenomenon: the divergence of the
Coulomb interactions that may be difficult to approximate and/or the mix
between attractive and repulsive contributions that prevents having a
variational character.

In this paper the two hypotheses have been investigated for the bosonic ground
state of systems of three identical particles. It has been shown that the
presence of a divergence is correlated with a decrease of accuracy for the ET
while a mix of variational character may result in a loss of efficacy of the
improved method. These two phenomena can together explain the unsatisfying
efficiency of the ET in the evaluation of atomic spectra. The divergence of
the Coulomb interactions appears to decrease the accuracy of the original
method while the mix of variational character prevents the improvement to
correct the obtained energies.

Even if the conclusions drawn in this paper bring an answer to the initial
question, this answer may still be incomplete. In the future, the above
conclusions may be confirmed for systems of $N_{a}$ identical particles plus
one different. The role of the particle masses (quite minor for identical
particles) and the excited states are other tracks that may be investigate for
a better understanding of the conditions in which the ET is accurate.

###### Acknowledgements.

L.C. and C.C would thank the Fonds de la Recherche Scientifique - FNRS for the
financial support. E.C. and J.V. would thank the University of Mons for their
Initiation Research Grant. This work was also supported under Grant Number
4.45.10.08. The authors thank Prof. Claude Semay for the careful reading of
the manuscript and the fruitful discussions.

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