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\begin{document}

\title{A Systematic Study of Structures, Stability, and Electronic Properties
of alloy clusters AlBe\textsubscript{\emph{n}} (\emph{n} = 1--12):
Comparison with Pure Beryllium Clusters}



\author[1]{Dan Yu}%
\author[2]{Wei-Ming Sun}%
\author[3]{Jing-yao Liu}%
\author[1]{Wu Di}%
\author[3]{Ying Li}%
\author[3]{Zhi-Ru Li}%
\affil[1]{Jilin University}%
\affil[2]{Department of Basic Chemistry, The School of Pharmacy, Fujian Medical University}%
\affil[3]{Institute of theoretical chemistry Jilin University}%


\vspace{-1em}



  
  \date{June 14, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
The geometric structures, energetic and electronic properties of global
minima of the AlBe\textsubscript{\emph{n}} (\emph{n} = 1--12) clusters
have been systemically studied by using the hybrid density functional
theory {[}B3LYP{]} and coupled cluster {[}CCSD(T){]} methods. It is
found that the impurity Al atom is externally bound to the host Ben
framework and its maximum coordination number is six. Besides, the
geometries of AlBe\emph{\textsubscript{n}} bear close resemblance to
either local or global minimum structures of
Be\emph{\textsubscript{n}}\textsubscript{+1}. The AlBe\textsubscript{3}
and AlBe\textsubscript{8} clusters exhibit high relative stability among
the AlBe\emph{\textsubscript{n}} clusters, which is reflected by the
evolutions of average atomic binding energy, dissociation energy, second
difference in energy, adsorption energy of Al, and HOMO-LUMO gap with
cluster size. In comparison to the pure
Be\emph{\textsubscript{n}}\textsubscript{+1} clusters,
AlBe\emph{\textsubscript{n}} exhibit larger binding energy values,
whereas they are more polarizable.%
\end{abstract}%



\sloppy


\textbf{A Systematic Study of Structures, Stability, and Electronic
Properties of alloy clusters AlBe\emph{\textsubscript{n}} (\emph{n} =
1--12): Comparison with Pure Beryllium Clusters}

\textbf{Dan Yu,} 11\emph{Institute of Theoretical Chemistry, Jilin
University, Changchun 130023, (P. R. China)}\textbf{Wei-Ming Sun,}
22\emph{The School of Pharmacy, Fujian Medical University, Fuzhou
350108, (P. R. China)} \textbf{Jing-Yao Liu,\textsuperscript{1} Di
Wu,\textsuperscript{1} Ying Li,\textsuperscript{1} and Zhi-Ru
Li\textsuperscript{1}}

Correspondence to: Wei-Ming Sun (E-mail:\emph{sunwm@fjmu.edu.cn}); Ying
Li (E-mail: \emph{liyingedu@jlu.edu.cn} )

\textbf{Introduction}

As a bridge between atoms and macroscopic state, cluster science
continues to be a subject of increasing research interest. In the realm
of cluster investigations, metal
clusters\textsuperscript{{[}1--7{]}}have attracted extensive attention
in the last several decades because they exhibit many intriguing
properties that are neither atomic-like nor extended
solidlike,\textsuperscript{{[}8,9{]}} while being closely connected to
size, geometry, and composition. These size-dependent properties offer
exciting possibilities for developing finely tuned cluster-assembled
materials, which have promising applications in the fields of material
science, optics, nanotechnology, catalysis,
etc.\textsuperscript{{[}10--14{]}}Besides, it has long been recognized
that the properties of metal clusters, instead of evolving linearly with
size, usually vary discontinuously with the number of component
atoms.\textsuperscript{{[}11{]}} In this respect, a simple yet helpful
free-electron model, namely spherical jellium model
(SJM),\textsuperscript{{[}15{]}} has been proposed to account for the
evolution of properties (even structures) of numerous pure and doped
metal clusters.\textsuperscript{{[}16,17{]}}

The fact that beryllium dimer is weakly
bound\textsuperscript{{[}18--22{]}} whereas bulk beryllium is a hard
metal with rather high melting point and enthalpy of
atomization\textsuperscript{{[}23{]}} makes beryllium cluster an ideal
prototype for exploring the evolution from discrete molecules to
metallic state, as indicated in an overview on the studies of beryllium
clusters.\textsuperscript{{[}24{]}} In contrast to experimental studies
which are limited by the toxicity of beryllium, quantum chemistry
provides powerful computational methods that help to investigate
beryllium clusters in detail. By using density functional theory with
Becke--Lee--Yang--Parr gradient correction, Wang et
al.\textsuperscript{{[}25{]}} studied the structural and electronic
properties of beryllium clusters containing up to 21 atoms, and they
found that the Be\textsubscript{4}, Be\textsubscript{10} and
Be\textsubscript{17} clusters show particularly high stability. With the
help of a modified genetic algorithm, Khanna and
coworkers\textsuperscript{{[}26{]}} have revealed the dependence of
relative stability and electronic properties of the
Be\emph{\textsubscript{n}} (\emph{n} = 2 - 41) clusters on their
equilibrium geometries.

Recently, doped metal clusters have attracted much attention from both
theoretical and experimental researchers. It has been found that the
characteristics of pure metal clusters, such as relative stability,
structural evolution, bonding character, electronic and magnetic
properties, etc., are usually altered when a heteroatom is introduced.
Consequently, doped clusters may show new physicochemical properties not
found in pure clusters. In this study, we focus on doped beryllium
clusters and choose Al atom as the dopant atom. On the one hand,
impurity-atom doping may provide additional flexibility to modulate the
physical and chemical properties of beryllium clusters. On the other
hand, the systematical study of beryllium-aluminum binary clusters is a
meaningful project in view of the important applications of Be-Al alloys
in disk drive armatures, automotive braking systems, and aerospace and
satellite system components.\textsuperscript{{[}27-30{]}}

In the present work, we have performed a comprehensive study of the
bimetallic AlBe\emph{\textsubscript{n}} (\emph{n} = 1--12) clusters.
Besides exploring the ground state structures, we also aim to reveal the
evolution of structures, stability, and various electronic structure
related properties of AlBe\emph{\textsubscript{n}} along with cluster
size. Furthermore, we make a comparison between
AlBe\emph{\textsubscript{n}} and pure Be\textsubscript{\emph{n}
+1}clusters from all aspects to analyze the Al-substitution effect. We
hope that results from this study will not only offer a fundamental
understanding of structure-property relationship of subnano-scale
beryllium-aluminum clusters, but provide useful references for studies
of other binary metal clusters.

\textbf{2. Methods}

The lowest-energy structures of the
AlBe\emph{\textsubscript{n}}(\emph{n} = 1--12) clusters were identified
by using two methods, for both of which the spin multiplicity of 2 and 4
were taken into account. In the first one, a large number of
AlBe\emph{\textsubscript{n}}geometries were manually built on the basis
of those of pure beryllium clusters.\textsuperscript{{[}16{]}} We have
obtained various structures where the impurity Al atom is either
attached to each possible site of the Be\emph{\textsubscript{n}} cluster
or used to substitute one Be atom of the host Be\textsubscript{\emph{n}
+1} cluster. In the second one, a stochastic search
procedure\textsuperscript{{[}2{]}}was used to make sure that all the
global minimum structures have been found. This procedure can generate
structures randomly, and hence facilitates a thorough exploration of
unknown isomers. With this method, a great number of starting geometries
were obtained at the B3LYP/LANL2DZ level. Afterwards, much more hunts
were performed in the region of these structures to do an intensive
search until no new minima appeared. Then, all the possible initial
structures obtained from these two methods were optimized with
subsequent frequency analysis at the B3LYP/aug-cc-pVDZ level to identify
the lowest-energy ones.

To examine the energetic and electronic properties of the
AlBe\emph{\textsubscript{n}} clusters, single-point calculations were
performed by using both B3LYP and CCSD(T) methods with the aug-cc-pVDZ
basis set. Natural bond orbital (NBO)
analysis\textsuperscript{{[}33{]}}was carried out at the
B3LYP/aug-cc-pVDZ level. At the same computational level, the
spherically averaged polarizability (\selectlanguage{greek}α\selectlanguage{english}) of AlBe\emph{\textsubscript{n}}
were calculated, which is defined as the mean value of diagonal terms of
the polarizability matrix, namely,

All calculations were carried out by using the GAUSSIAN
09\textsuperscript{{[}34{]}} program package. Dimensional diagrams of
the molecular structures and orbitals were generated with the GaussView
program.\textsuperscript{{[}35{]}}

\textbf{RESULTS AND DISCUSSION}

3.1 Geometrical structures

First, we focus on the geometrical evolution of the
AlBe\emph{\textsubscript{n}}(\emph{n} = 1--12) clusters. To explore the
Al substitution effect on beryllium clusters, the structures of
Be\textsubscript{\emph{n} +1} were also obtained on the basis of
previous reports\textsuperscript{{[}25,26,36{]}} and reoptimized at the
B3LYP/aug-cc-pVDZ level. The lowest-energy structures of
AlBe\emph{\textsubscript{n}} along with the corresponding global and
local minima of the Be\textsubscript{\emph{n} +1} (\emph{n} = 1--12)
clusters are presented in Figure 1, where their spin multiplicity are
also shown. It has been found that all the global minima of
AlBe\emph{\textsubscript{n}} are in doublet states except for
AlBe\textsubscript{7} corresponding to a quartet state. The symmetry,
number of Al--Be bonds, the lowest vibrational frequencies, the shortest
Al-Be bond lengths, as well as the HOMO--LUMO gaps of the most stable
AlBe\emph{\textsubscript{n}} (\emph{n} = 1--12) clusters are listed in
Table 1.

The ground state structure of AlBe has a bond length of 2.439 \selectlanguage{ngerman}Å and
vibrational frequency of 374 cm\textsuperscript{--1}, according well
with experimental results (\emph{R} = 2.395 Å, \emph{v} \emph{=} 382
cm\textsuperscript{--1})\textsuperscript{{[}37{]}}and previously
reported theoretical results (\emph{R}= 2.430 Å, \emph{v} \emph{=} 385
cm\textsuperscript{--1};\textsuperscript{37} \emph{R} = 2.423 Å,\emph{v}
\emph{=} 375 cm\textsuperscript{--1};\textsuperscript{{[}38{]}}\emph{R}
= 2.434 Å, \emph{v} \emph{=} 379.6
cm\textsuperscript{--1};\textsuperscript{{[}39{]}}). Thus, the
B3LYP/aug-cc-pVDZ treatment is considered reliable for predicting the
structures of the AlBe\emph{\textsubscript{n}} clusters. The global
minimum of AlBe\textsubscript{2} exhibits an isosceles triangular
structure with \emph{C} \textsubscript{2\emph{v}} symmetry, in which the
Al--Be and Be--Be bond lengths are 2.302 Å and 2.153 Å, respectively.
Note that the Be-Be bond length of AlBe\textsubscript{2}is shorter by
0.351 Å than that of Be\textsubscript{2}, indicating that the Be--Be
bond strength of Be\textsubscript{2} has been enhanced by the attachment
of Al atom. Likewise, the Al-Be bond of AlBe\textsubscript{2}is also
shorter than that of AlBe.

The transition from two-dimensional (2D) to three-dimensional (3D)
structures occurs at \emph{n} = 3 in this series. From Figure 1, a
tetrahedral configuration (\emph{C} \textsubscript{3\emph{v}} ) is
obtained for AlBe\textsubscript{3}, where the Al atom caps a equilateral
trianglar Be\textsubscript{3} unit. In this structure, the Al--Be and
Be--Be bond lengths are 2.324 Å and 2.048 Å, respectively. The latter is
just equal to that of the lowest-energy structure of
Be\textsubscript{4}. In the case of AlBe\textsubscript{4}, the most
stable structure is a \emph{C\textsubscript{s}} -symmetric trigonal
bipyramid with the Al atom at the top site. This structure contains two
short Al--Be bonds of 2.339 Å and a long Al--Be bond of 2.721 Å.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 1.} Ground-state geometries of the host
Be\textsubscript{\emph{n} +1}clusters (1 \textbf{[?]} \emph{n} \textbf{[?]}
12) on the left side. The lowest-energy structures of the
AlBe\emph{\textsubscript{n}} clusters (1\textbf{[?]} \emph{n} \textbf{[?]}
12) on the right. In the bottom, the global minimum structures of
Be\textsubscript{\emph{n} +1} (\emph{n} = 5, 6, 11, and 12) are shown.
Be\emph{\textsubscript{n}} -g represents a global minimum and the
Be\emph{\textsubscript{n}} -l represents a local minimum and the
multiplicity of each structure is displayed in parentheses. Be atoms are
shown as yellow spheres while Al atoms in pink color.

It can be noticed that, from AlBe to AlBe\textsubscript{4}, the larger
structure can be obtained by attaching a Be atom to the smaller one. As
for AlBe\textsubscript{5}, the lowest-energy structure prefers
a\emph{C\textsubscript{s}} -symmetric cap-like configuration, which
looks similar to a metastable structure of
Be\textsubscript{6}(Be\textsubscript{6}-\emph{l} in Figure 1). The
Al-Be1 and Al-Be2 bond lengths of AlBe\textsubscript{5} are 2.610 \selectlanguage{ngerman}Å and
2.274 Å, respectively, and the Be--Be bond lengths vary from 1.992 Å to
2.167 Å. The lowest-lying state of AlBe\textsubscript{6} has a
pentagonal bipyramidal geometry of \emph{C} \textsubscript{2} symmetry.
It can be obtained by attaching a Be atom to AlBe\textsubscript{5} from
the opposite side of the Be1 atom, which accordingly shortens the Al-Be1
bond by 0.127 Å. Just like the case of AlBe\textsubscript{5}, the ground
state of AlBe\textsubscript{6} has a similar configuration to that of a
local, but not global, minimum of Be\textsubscript{7}. The most stable
structure of AlBe\textsubscript{7} presents a bicapped octahedral
structure, which can be obtained by either duplicating the Be2 atom of
AlBe\textsubscript{6} or substituting an Al for the top Be atom in the
global minimum of Be\textsubscript{8} (Be\textsubscript{8}-\emph{g} ).
Note that a similar structure with a spin multiplicity of 2 was also
obtained for AlBe\textsubscript{7}, whereas it is 4.47 kcal/mol less
stable than the quartet state.

As can be seen from Figure 1, each structure of the
AlBe\emph{\textsubscript{n}} (\emph{n} = 8-12) series could be generated
by attaching a Be atom to that of AlBe\textsubscript{\emph{n} -1}. The
lowest-energy structure of AlBe\textsubscript{8} can also be regarded as
a result of substituting an Al atom for a Be atom in
Be\textsubscript{9}-\emph{g} . A 16-faced deltahedron was identified to
be the most stable structure of AlBe\textsubscript{9}. Obviously, this
structure can be obtained by an additional Be atom face-capping the
bottom of AlBe\textsubscript{8}. Meanwhile, it quite resembles the
global minimum structure of Be\textsubscript{10}. The ground-state
structure of AlBe\textsubscript{10} bears strong resemblance to that of
Be\textsubscript{11}. It can be considered derived from
AlBe\textsubscript{9} by twinning the top Be1 atom of the latter. The
global minimum of AlBe\textsubscript{11} presents a capsule geometry
derived from a local minimum structure (Be\textsubscript{12}-\emph{l} )
of Be\textsubscript{12} instead of the icosahedral
Be\textsubscript{12}-\emph{g} . On the other hand, face-capping the
AlBe\textsubscript{10}polyhedron can also generate the structure of
AlBe\textsubscript{11}. Meanwhile, the introduction of the Be5 atom
leads to lengthened Be1--Be3 distance (from 2.095 Å to 3.102 Å), so the
Be1--Be3 bond is broken in AlBe\textsubscript{11}. The most stable
structure of AlBe\textsubscript{12} can be regarded as the result of
attaching a Be (Be8) atom to AlBe\textsubscript{11}, accompanied by
broken Be4-Be6 and Be4-Be7 bonds. In this group of structures, the
Al--Be bond lengths range from 2.285 Å to 2.505 Å and the Be--Be bond
lengths are 2.044--2.220 Å.

\textbf{Table 1.} Symmetry Point Groups, the Number of Be--Al Bonds (N),
the Shortest Be--Al Bond Lengths (\emph{R} \textsubscript{Be-Al}, in Å),
the Lowest Vibrational Frequencies (\emph{v} , in
cm\textsuperscript{-1}), and the HOMO--LUMO Gaps (in eV) of the
Lowest-energy AlBe\emph{\textsubscript{n}} (\emph{n} = 1--12) Clusters.\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
Isomers & Symmetry & N & \emph{R}\textsubscript{Be-Al} & \emph{v} &
gap\tabularnewline
AlBe & \emph{C\textsubscript{[?]v}} & 1 & 2.439 & 374 &
1.398\tabularnewline
AlBe\textsubscript{2} & \emph{C}\textsubscript{2\emph{v}} & 2 & 2.302 &
415 & 1.714\tabularnewline
AlBe\textsubscript{3} & \emph{C}\textsubscript{3\emph{v}} & 3 & 2.324 &
376 & 2.194\tabularnewline
AlBe\textsubscript{4} & \emph{C\textsubscript{s}} & 3 & 2.339 & 87 &
1.661\tabularnewline
AlBe\textsubscript{5} & \emph{C\textsubscript{s}} & 3 & 2.274 & 113 &
2.062\tabularnewline
AlBe\textsubscript{6} & \emph{C}\textsubscript{2} & 4 & 2.289 & 67 &
1.482\tabularnewline
AlBe\textsubscript{7} & \emph{C\textsubscript{s}} & 5 & 2.323 & 127 &
2.514\tabularnewline
AlBe\textsubscript{8} & \emph{C\textsubscript{s}} & 5 & 2.285 & 212 &
2.817\tabularnewline
AlBe\textsubscript{9} & \emph{C}\textsubscript{3\emph{v}} & 6 & 2.362 &
215 & 2.324\tabularnewline
AlBe\textsubscript{10} & \emph{C}\textsubscript{2} & 6 & 2.344 & 215 &
2.036\tabularnewline
AlBe\textsubscript{11} & \emph{C}\textsubscript{1} & 6 & 2.309 & 203 &
1.758\tabularnewline
AlBe\textsubscript{12} & \emph{C}\textsubscript{1} & 6 & 2.306 & 187 &
1.754\tabularnewline
\bottomrule
\end{longtable}

As discussed above, there is a counterpart for each lowest-energy
structure of AlBe\emph{\textsubscript{n}} in the minimum structures of
their corresponding Be\textsubscript{\emph{n} +1} cluster. Besides, as
can be seen in Table 1, the number of Al--Be bonds in
AlBe\emph{\textsubscript{n}} increases with the increasing cluster size
and reaches a maximum value of six. It implies that the impurity Al atom
is able to bond with six Be atoms at most. These rules may help to
identify the low-lying structures of larger Al-doped
Be\emph{\textsubscript{n}} clusters.

The comparison between AlBe\emph{\textsubscript{n}} and other Al-doped
clusters shows how the relative size of impurity atom affects the
structural evolution of the whole system. For example, the impurity Al
atom always occupies an external position of the host clusters in
AlBe\emph{\textsubscript{n}} and AlB\emph{\textsubscript{n}}
\textsuperscript{{[}40{]}} until \emph{n} = 12, while it gets trapped in
the host cage from \emph{n} = 6 onwards in AlLi\emph{\textsubscript{n}}
\textsuperscript{{[}41{]}} and AlNa\emph{\textsubscript{n}}
,\textsuperscript{{[}42,43{]}} and from\emph{n} = 9 onwards in the
AlTi\emph{\textsubscript{n}}
,\textsuperscript{{[}44{]}}AlSc\emph{\textsubscript{n}}
,\textsuperscript{{[}45{]}}AlPb\emph{\textsubscript{n}}
,\textsuperscript{{[}46{]}} and AlY\emph{\textsubscript{n}}
\textsuperscript{{[}47{]}} clusters. These structural distinctions can
be related to the size difference of the dopant atom versus the host
atom. To be specific, the impurity Al atom (1.25
\selectlanguage{ngerman}Å)\textsuperscript{{[}48{]}} has larger atomic radius than those of B
(0.85 Å) and Be (1.05 Å) atoms, which hinders it from entering the
B\emph{\textsubscript{n}} or Be\emph{\textsubscript{n}} cages. In
contrast, a framework constituted by larger host atoms such as Li (1.45
Å), Na (1.80 Å), Ti (1.40 Å), Sc (1.60 Å), Pb (1.80) and Y (1.80 Å) can
accommodate the dopant Al atom.

3.2 Stability and Electronic Properties

\emph{3.2.1 Stability}

The relative stability of different sized
AlBe\emph{\textsubscript{n}}clusters can be discussed on the basis of
binding energy per atom (\emph{E\textsubscript{b}} ), dissociation
energy (\selectlanguage{greek}Δ\selectlanguage{english}\emph{E} ), and the second difference in energy
(\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}\emph{E} ), where

These energetic properties, calculated at the CCSD(T)//B3LYP and B3LYP
levels, are listed in Table S1 in supporting information. From the
table, the B3LYP method overestimates binding energies and dissociation
energies by 0.103--0.254 eV and 0.147--0.387 eV, respectively, when
compared with the CCSD(T) results. Nevertheless,
the\emph{E\textsubscript{b}} and \selectlanguage{greek}Δ\selectlanguage{english}\emph{E} values obtained by both
methods show consistent trends. Besides, the B3LYP results of
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}\emph{E} accord well with those at the CCSD(T)
level, and the differences are 0.033 eV--0.240 eV.

To explore the Al-substitution effect on the stability of bare beryllium
clusters, the above-mentioned quantities of Be\textsubscript{\emph{n}
+1}were also calculated according to the following equations,

For comparison, the size dependence of these energetic properties for
the lowest-energy AlBe\emph{\textsubscript{n}} and
Be\textsubscript{\emph{n+} 1} (\emph{n} = 1--12) clusters are plotted in
Figure 2. It is known that small bumps and dips in
the\emph{E\textsubscript{b}} curve are indicative of relative stability
and reactivity for corresponding clusters, respectively. From Figure 2a,
the\emph{E} \textsubscript{b} values of AlBe\emph{\textsubscript{n}} and
Be\textsubscript{\emph{n} +1} increase sharply first, then, both curves
reach a plateau from \emph{n} = 3 to \emph{n} = 5 and then rise again as
cluster size grows. It is noted that the \emph{E\textsubscript{b}} value
of an AlBe\emph{\textsubscript{n}} cluster is larger by 0.033--0.324 eV
than that of its corresponding Be\textsubscript{\emph{n} +1} cluster,
suggesting that the substitution of an Al arom for a Be atom in
Be\textsubscript{\emph{n} +1} can enhance the intracluster binding
force.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

\textbf{Figure 2.} Size dependence of binding energies per atom
(\emph{E\textsubscript{b}} ), dissociation energies (\selectlanguage{greek}Δ\selectlanguage{english}\emph{E} ), and
the second difference in energy (\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}\emph{E} ) for the
lowest-energy AlBe\emph{\textsubscript{n}} and
Be\textsubscript{\emph{n+} 1} (\emph{n} = 1--12) clusters at the
CCSD(T)/aug-cc-pVDZ level.

In addition, the \emph{E\textsubscript{b}} curve of
AlBe\emph{\textsubscript{n}} shows two turning points at \emph{n} = 3
and \emph{n} = 8, indicating relatively high stability of
AlBe\textsubscript{3} and AlBe\textsubscript{8}. This is more clearly
reflected in varying trends of \selectlanguage{greek}Δ\selectlanguage{english}\emph{E} and
\selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}\emph{E} . As shown in Figure 2b, the \selectlanguage{greek}Δ\selectlanguage{english}\emph{E}
curve of AlBe\emph{\textsubscript{n}} has two distinct peaks at
AlBe\textsubscript{3} and AlBe\textsubscript{8}, indicating that these
two clusters are less likely to lose a Be atom. As for
Be\textsubscript{\emph{n} +1}, two local maximum \selectlanguage{greek}Δ\selectlanguage{english}\emph{E} values appear
at Be\textsubscript{4} and Be\textsubscript{10} because they are magic
clusters and possess unique stability according to spherical jellium
model (SJM)\textsuperscript{\emph{15}} . A similar situation can be
found in the evolution of \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}\emph{E} values. From
Figure 2c, the curves of \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}\emph{E} have particular
peaks at \emph{n} = 3 and\emph{n} = 8 for AlBe\emph{\textsubscript{n}} ,
and at \emph{n} = 3 and\emph{n} = 9 for the Be\textsubscript{\emph{n}
+1} series. Combining all the results given above, it can be concluded
that AlBe\textsubscript{3}and AlBe\textsubscript{8} have special
stability among the AlBe\emph{\textsubscript{n}} clusters as do
Be\textsubscript{4} and Be\textsubscript{10} in the
Be\textsubscript{\emph{n} +1} system.

The stability of the AlBe\emph{\textsubscript{n}} clusters is also
examined in terms of adsorption energy of Al, \emph{i.e.} , the energy
released when an Al atom is attached to a pure
Be\emph{\textsubscript{n}} cluster, which can be expressed as

The \emph{E} \textsubscript{ad} values of
AlBe\emph{\textsubscript{n}}were also obtained by using the
CCSD(T)//B3LYP and B3LYP methods, and the results are collected in Table
S1 and plotted in Figure 3. From the figure, the two methods yield
consistent results, and there are two obvious dips at
AlBe\textsubscript{3} and AlBe\textsubscript{8} in both curves. These
dips denote that a lot of energy is released when an Al atom is adsorbed
by either Be\textsubscript{3} or Be\textsubscript{8}cluster. From Table
S1, all the \emph{E} \textsubscript{ad} values are negative, indicating
that the adsoption of Al on Be\emph{\textsubscript{n}} is a favorable
process. Besides, AlBe\textsubscript{8} exhibits the largest \emph{E}
\textsubscript{ad}value (-4.132 eV at CCSD(T) level and -3.872 eV at
B3LYP level), so the impurity Al atom is tightly bound to the host
Be\textsubscript{8}cluster.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}

\end{center}
\end{figure}

\textbf{Figure 3.} Size dependence of adsorption energies of Al
(\emph{E} \textsubscript{ad}) for the lowest-energy
AlBe\emph{\textsubscript{n}} (\emph{n} = 1--12) clusters at both
CCSD(T)/aug-cc-pVDZ and B3LYP/aug-cc-pVDZ levels.

\emph{3.2.2 Electronic Properties}

The stability of clusters can also be evaluated by certain electronic
structure related descriptors including vertical ionization potential
(\emph{VIP} ), vertical electron affinity (\emph{VEA} ), chemical
hardness, the highest occupied--lowest unoccupied molecular orbital
(HOMO--LUMO) energy gap, and polarizability, etc. \emph{VIP} is the
energy difference between the ground state of the neutral cluster and
the ionized cluster that has the same geometry as the neutral. In this
work, the \emph{VIP} s of the lowest-energy structures of
AlBe\emph{\textsubscript{n}} were calculated at the B3LYP and CCSD(T)
levels. From the results shown in Figure S1a, the two curves show
generally consistent trends. Either curve has two striking peaks
at\emph{n} = 2 and 8, implying that the AlBe\textsubscript{2} and
AlBe\textsubscript{8} clusters are not apt to lose an electron.
Meanwhile, the CCSD(T) results of AlBe\emph{\textsubscript{n}} and
Be\textsubscript{\emph{n} +1} clusters are compared in Table 2. From the
table, the \emph{VIP} value of AlBe\textsubscript{\emph{n}} is always
smaller than that of its corresponding Be\textsubscript{\emph{n}
+1}cluster except for \emph{n} = 6.

\textbf{Table 2.} Vertical Ionization Potential (\emph{VIP} , in eV),
Vertical Electron Affinity (\emph{VEA} , in eV), and Hardness (\selectlanguage{greek}\emph{η}
, \selectlanguage{english}in eV) of the Lowest-energy Be\textsubscript{\emph{n} +1} and
AlBe\emph{\textsubscript{n}} (\emph{n} = 1--12) Clusters at the
CCSD(T)// B3LYP/aug-cc-pVDZ Level.\selectlanguage{english}
\begin{longtable}[]{@{}l@{}}
\toprule
\begin{minipage}[b]{0.97\columnwidth}\raggedright\strut
3.572\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.97\columnwidth}\raggedright\strut
\emph{n}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.97\columnwidth}\raggedright\strut
\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.97\columnwidth}\raggedright\strut
1\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

\&\& 5.995 \& 0.586 \&\selectlanguage{english}
\begin{longtable}[]{@{}l@{}}
\toprule
2.705\tabularnewline
\midrule
\endhead
2.683\tabularnewline
2.338\tabularnewline
2.516\tabularnewline
2.169\tabularnewline
2.341\tabularnewline
2.288\tabularnewline
2.159\tabularnewline
2.293\tabularnewline
2.156\tabularnewline
2.032\tabularnewline
1.889\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
\emph{n} & \emph{Q}\textsubscript{Al} & NEC(Al) & NEC(Be)\tabularnewline
1 & -0.167 &
{[}core{]}3\emph{s}\textsuperscript{1.96}3\emph{p}\textsuperscript{1.18}3\emph{d}\textsuperscript{0.03}
&
{[}core{]}2\emph{s}\textsuperscript{1.61}2\emph{p}\textsuperscript{0.22}\tabularnewline
2 & -0.621 &
{[}core{]}3\emph{s}\textsuperscript{1.90}3\emph{p}\textsuperscript{1.66}3\emph{d}\textsuperscript{0.05}
&
{[}core{]}2\emph{s}\textsuperscript{1.37}2\emph{p}\textsuperscript{0.31}\tabularnewline
3 & -0.821 &
{[}core{]}3\emph{s}\textsuperscript{1.89}3\emph{p}\textsuperscript{1.87}4\emph{s}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.05}
&
{[}core{]}2\emph{s}\textsuperscript{1.33}2\emph{p}\textsuperscript{0.37}3\emph{s}\textsuperscript{0.02}3\emph{p}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.01}\tabularnewline
4 & -0.658 &
{[}core{]}3\emph{s}\textsuperscript{1.83}3\emph{p}\textsuperscript{1.77}3\emph{d}\textsuperscript{0.05}
&
{[}core{]}2\emph{s}\textsuperscript{1.30-1.43}2\emph{p}\textsuperscript{0.37-0.53}3\emph{p}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.01}\tabularnewline
5 & -1.044 &
{[}core{]}3\emph{s}\textsuperscript{1.79}3\emph{p}\textsuperscript{2.18}3\emph{d}\textsuperscript{0.07}
&
{[}core{]}2\emph{s}\textsuperscript{1.18-1.45}2\emph{p}\textsuperscript{0.37-0.56}3\emph{s}\textsuperscript{0.01}3\emph{p}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.01-0.07}\tabularnewline
6 & -1.435 &
{[}core{]}3\emph{s}\textsuperscript{1.76}3\emph{p}\textsuperscript{2.56}3\emph{d}\textsuperscript{0.10}
&
{[}core{]}2\emph{s}\textsuperscript{1.15-1.23}2\emph{p}\textsuperscript{0.34-0.70}3\emph{p}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.01-0.10}\tabularnewline
7 & -1.306 &
{[}core{]}3\emph{s}\textsuperscript{1.79}3\emph{p}\textsuperscript{2.40}4\emph{s}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.10}
&
{[}core{]}2\emph{s}\textsuperscript{1.20-1.25}2\emph{p}\textsuperscript{0.49-0.68}3\emph{p}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.02-0.10}\tabularnewline
8 & -1.679 &
{[}core{]}3\emph{s}\textsuperscript{1.76}3\emph{p}\textsuperscript{2.78}4\emph{s}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.12}
&
{[}core{]}2\emph{s}\textsuperscript{1.18-1.31}2\emph{p}\textsuperscript{0.42-0.68}3\emph{p}\textsuperscript{0.01}3\emph{d}\textsuperscript{0.02-0.04}\tabularnewline
9 & -2.285 &
{[}core{]}3\emph{s}\textsuperscript{1.78}3\emph{p}\textsuperscript{3.30}4\emph{s}\textsuperscript{0.05}3\emph{d}\textsuperscript{0.15}
&
{[}core{]}2\emph{s}\textsuperscript{1.13-1.27}2\emph{p}\textsuperscript{0.43-0.59}3\emph{p}\textsuperscript{0.01-0.03}3\emph{d}\textsuperscript{0.01-0.05}\tabularnewline
10 & -2.403 &
{[}core{]}3\emph{s}\textsuperscript{1.73}3\emph{p}\textsuperscript{3.48}4\emph{s}\textsuperscript{0.04}3\emph{d}\textsuperscript{0.15}
&
{[}core{]}2\emph{s}\textsuperscript{1.07-1.28}2\emph{p}\textsuperscript{0.50-0.64}3\emph{p}\textsuperscript{0.01-0.03}3\emph{d}\textsuperscript{0.02-0.05}\tabularnewline
11 & -2.336 &
{[}core{]}3\emph{s}\textsuperscript{1.71}3\emph{p}\textsuperscript{3.41}4\emph{s}\textsuperscript{0.03}3\emph{d}\textsuperscript{0.17}
&
{[}core{]}2\emph{s}\textsuperscript{1.10-1.34}2\emph{p}\textsuperscript{0.34-0.71}3\emph{p}\textsuperscript{0.01-0.03}3\emph{d}\textsuperscript{0.02-0.06}\tabularnewline
12 & -2.331 &
{[}core{]}3\emph{s}\textsuperscript{1.69}3\emph{p}\textsuperscript{3.42}4\emph{s}\textsuperscript{0.03}3\emph{d}\textsuperscript{0.18}
&
{[}core{]}2\emph{s}\textsuperscript{1.07-1.37}2\emph{p}\textsuperscript{0.31-0.76}3\emph{p}\textsuperscript{0.01-0.03}3\emph{d}\textsuperscript{0.02-0.05}\tabularnewline
\bottomrule
\end{longtable}

can be found from Figure S2 that, the Al charge increases as the size of
cluster grows and tends to flatten out from
AlBe\textsubscript{9}onwards. Note that this varying trend roughly
corresponds to that of coordination number of Al in the
AlBe\emph{\textsubscript{n}} clusters, suggesting that higher
coordination number of impurity atom is beneficial to intracluster
charge transfer in this case. According to Table 3, the 3s states of Al
lose 0.04--0.31 electrons, while the 3p states get 0.18--2.48 electrons.
As for the Be atoms, their 2s orbitals lose 0.39--0.93 electrons, while
the 2p states get 0.22--0.76 electrons. Hence, the charge transfer
within the AlBe\emph{\textsubscript{n}}clusters is mainly from the 3s
orbital of Al and 2s orbitals of Be atoms to the 3p orbital of Al and 2p
orbitals of Be atoms. By contrast, the contributions from other orbitals
of Al and Be are negligible.

The HOMO-LUMO gaps of the AlBe\emph{\textsubscript{n}} clusters at the
B3LYP level are listed in Table 1 and plotted in Figure S3. From the
figure, there are three obvious peaks at AlBe\textsubscript{3},
AlBe\textsubscript{5} and AlBe\textsubscript{8}, indicating relatively
high stability of these three clusters. In particular, the
AlBe\textsubscript{8} cluster possesses the largest HOMO--LUMO gap of
2.817 eV, which not only is larger than the experimental gap value of
1.9 eV for the kinetically stable
C\textsubscript{60},\textsuperscript{{[}47{]}} but also exceeds that of
2.53 eV (computed at the same level) for the chemically inert superatom
Al\textsubscript{13}\textsuperscript{--}.\textsuperscript{{[}48{]}} For
comparison, the HOMO--LUMO gaps of the Be\textsubscript{\emph{n}
+1}clusters are also depicted in Figure S3. It can be found that
substituting an Al atom for a Be atom enlarges the HOMO--LUMO gaps of
Be\textsubscript{8} and Be\textsubscript{10} clusters by 0.689 eV and
0.398 eV, respectively, while the Al substitution effect is
insignificant for Be\textsubscript{11} and Be\textsubscript{13} clusters
and is negative for the other beryllium clusters.

Finally, the evolution of polarizability per atom of
AlBe\textsubscript{\emph{n}} (\emph{n} = 1--12) is considered since the
static polarizability is an important measure of electronic properties
of clusters. From the results shown in Figure 4, a turning point
at\emph{n} = 3 can be found, which squares with the geometry
transformation from planar to steroescopic at AlBe\textsubscript{3}. For
comparison, the polarizability per atom of the lowest-energy
Be\textsubscript{\emph{n} +1} clusters are also given in Figure 4. From
the figure, both curves show generally decreasing trends with increasing
cluster size, and gradually become stable for larger sizes. Besides,
substituting a Be atom in Be\textsubscript{\emph{n} +1} with an Al atom
always brings about a larger polarizability to the doped cluster,
although the Al-substitution effect is less prominent for larger-sized
ones. Hence, the AlBe\emph{\textsubscript{n}} clusters are more
polarizable compared with their corresponding Be\textsubscript{\emph{n}
+1} clusters. Note that there is a relatively large polarizability gap
between Be\textsubscript{6} and AlBe\textsubscript{5}, which can be
attributed to the large structural discrepancy between these two
clusters since polarizability is sensitive to the shape of the
system.\textsuperscript{{[}50{]}}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image14/image14}

\end{center}
\end{figure}

\textbf{Figure 4.} Size dependence of polarizabilities per atom for the
lowest-energy AlBe\emph{\textsubscript{n}} and
Be\textsubscript{\emph{n+} 1} (\emph{n} = 1--12) clusters at the
B3LYP/aug-cc-pVDZ level.

3.3 In-depth study of AlBe\textsubscript{3} and AlBe\textsubscript{8}

Based on the preceding analysis of energetic and electronic properties,
it is interesting to find that the AlBe\textsubscript{3} and
AlBe\textsubscript{8} clusters possess special stability among the
AlBe\emph{\textsubscript{n}} series. Hence, the electronic structure and
density of states (DOS) of these two clusters are further investigated.
To facilitate comparison, the total and partial density of states (TDOS
and PDOS) of AlBe\textsubscript{3} and AlBe\textsubscript{8} as well as
their corresponding Be\textsubscript{4} and Be\textsubscript{9} clusters
are plotted as a function of energy in Figure 5, in which their valence
molecular orbitals (MOs) obtained at the ROMP2//B3LYP level are also
illustrated.

From Figure 5b, the singly-occupied HOMO orbital of
AlBe\textsubscript{3} exhibits dominant \emph{s} state, the HOMO-1 to
HOMO-3 orbitals present typical \emph{p} shape, and the HOMO-4 orbital
shows spherically symmetric \emph{s} character. So the valence electron
configuration of AlBe\textsubscript{3} is\emph{1S}
\textsuperscript{2}\emph{1P} \textsuperscript{6}\emph{2S}
\textsuperscript{1}. As for the AlBe\textsubscript{8} cluster, its
nineteen valence electrons are arranged in the\emph{1S}
\textsuperscript{2}\emph{1P} \textsuperscript{6}\emph{1D}
\textsuperscript{10}\emph{2S} \textsuperscript{1}pattern, which accords
well with the prediction of the spherical jellium
model.\textsuperscript{{[}15{]}} Compared with AlBe\textsubscript{8},
the triplet Be\textsubscript{9} cluster is one D electron less and has a
electronic shell structure of\emph{1S} \textsuperscript{2}\emph{1P}
\textsuperscript{6}\emph{1D} \textsuperscript{9}\emph{2S}
\textsuperscript{1}(see Figure 5c). As the studied
AlBe\emph{\textsubscript{n}} clusters are all open-shell species, there
are not any shell-closed magic clusters in this series. However, the
AlBe\textsubscript{3} and AlBe\textsubscript{8} clusters do exhibit
enhanced stability relative to their congeners. A possible explanation
is that both of them are one electron short of shell closure, and their
desire for an additional electron makes the component atoms bind so
tightly with each other that the clusters are stable against
dissociation of any kind.

From Figure 5, the introduction of Al impurity has more effect on the
DOS of p-type molecular orbitals of beryllium clusters. The comparison
between TDOS of Be\textsubscript{4} and AlBe\textsubscript{3} shows that
the peak near -6 eV for the former splits into two peaks in the latter.
Similarly, the peak around -9.5 eV in the TDOS plot of
Be\textsubscript{9} also splits into two peaks at -9.5 eV and -10 eV in
that of AlBe\textsubscript{8}. In contrast, the Al doping effect is
minor on the DOS of other states.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image15/image15}

\end{center}
\end{figure}

\textbf{Figure 5.} Density of states (DOS) and valence molecular
orbitals (shown as insets) of the AlBe\textsubscript{3},
AlBe\textsubscript{8}, Be\textsubscript{4}, and
Be\textsubscript{9}clusters.

\textbf{Conclusions}

The structures, relative stability, and electronic properties of the
lowest-energy AlBe\emph{\textsubscript{n}} (\emph{n} = 1--12) clusters
were systematically studied by high-level \emph{ab initio} and density
functional theories. The geometry transformation from planar to
steroescopic occurs at AlBe\textsubscript{3} in this series. The
impurity Al atom prefers to reside ouside the host
Be\textsubscript{\emph{n}} cluster until \emph{n} = 12 and only slightly
affects the configuration of the latter. NBO analysis reveals that
0.167\textasciitilde{}2.403\textbar{}e\textbar{} charge flows from the
host Be\emph{\textsubscript{n}} to the Al atom, which might be due to
the larger electronegativity of Al than Be. The evolutions
of\emph{E\textsubscript{b}} , \selectlanguage{greek}Δ\selectlanguage{english}\textsuperscript{2}\emph{E} , \selectlanguage{greek}Δ\selectlanguage{english}\emph{E}
,\emph{E} \textsubscript{ad}, and HOMO-LUMO gap with cluster size show
special stability of AlBe\textsubscript{3} and
AlBe\textsubscript{8}among the AlBe\emph{\textsubscript{n}} clusters.
Moreover, a comparison has been made between
AlBe\emph{\textsubscript{n}} and pure beryllium clusters, and it is
found that substituting an Al atom for a Be atom in
Be\textsubscript{\emph{n} +1} results in enhanced binding energy and
polarizability but decreased chemical hardness of the system.

\textbf{Acknowledgments}

This work was supported by the National Natural Science Foundation of
China (Grant No. 21573089, 51872057, 21603032), the ''13th Five-Year''
Science and Technology Research Project of Jilin provincial education
department (JJKH20190117KJ), Natural Science Foundation of Fujian
Province (Grant No. 2016J05032, 2016J01771), and MiaoPu
Foundation~of~Fujian~Medical~University (2015MP034).

\textbf{Keywords:}doped cluster, Be-Al alloy, bimetallic, optimization,
stability

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