Al doped ge 2015

Computational and Theoretical Chemistry 1054 (2015) 8–15

Contents lists available at ScienceDirect

Computational and Theoretical Chemistry
journal homepage: www.elsevier.com/locate/comptc

A computational investigation of aluminum-doped germanium clusters
by density functional theory study
Shunping Shi a,⇑, Yiliang Liu b, Chuanyu Zhang a, Banglin Deng a, Gang Jiang c
a

Department of Applied Physics, Chengdu University of Technology, Chengdu 610059, China
College of Electrical and Information Engineering, Southwest University for Nationalities, Chengdu 610041, China
c
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
b

a r t i c l e

i n f o

Article history:
Received 9 September 2014
Received in revised form 27 October 2014
Accepted 4 December 2014
Available online 11 December 2014
Keywords:
Density functional theory
Gen+1 cluster
GenAl clusters

Structure of clusters

a b s t r a c t
We report a computational study of the aluminum doped germanium clusters GenAl (n = 1–9). The
molecular geometries and electronic structures of the GenAl clusters are investigated systematically using
quantum calculations at the B3LYP level with the 6-311G(d) basis sets. The growth pattern behaviors, stabilities, electronic properties, and magnetic moments of these clusters are discussed in detail. Obviously
different growth patterns appear between small and larger Al-doped germanium clusters, the optimized
equilibrium geometries trend to prefer the close-packed configurations for Al-doped germanium clusters
up to n = 9. The size dependence of cluster average binding energies per atom (Eb/atom), second-order
differences of total energies (D2E), fragmentation energies (Ef) and HOMO–LUMO gaps of Gen+1 and GenAl
(n = 1–9) clusters are studied. The stability results show that Gen+1 cluster possess relatively higher
stability than GenAl cluster. Furthermore, the investigated highest occupied molecular orbital-lowest
unoccupied molecular orbital gaps indicate that the Gen+1 and GenAl clusters have different HOMO–
LUMO gap. In addition, the calculated vertical ionization potentials and vertical electron affinities confirm
the electric properties of Gen+1 and GenAl clusters. Besides, the doping of Al atom also brings the decrease
as the cluster sizes increase for atomic magnetic moments (lb).
Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction
The semiconductor clusters with transition metal have
attracted great interest for optoelectronic materials, catalyst, and
the development of new species in nanoscale applications. Germanium clusters have also widely been studied because they are
important for the fine processing of semiconductors and the synthesis of novel materials. The studies have shown that the structure and the bonding of bulk germanium are very similar to that
of bulk silicon, and the bulk surfaces show similar reconstruction
[1]. However, although small silicon and germanium clusters
appear to have similar geometries, the larger ones are fundamentally different [2]. During the past two decades, Gen clusters have
been intensively studied both experimentally [3–9] and theoretically [10–21] because of their fundamental importance and potential applications in nanoelectronics. The photoionization study has
been investigated by Yoshida and Fuke to characterize the electronic structures of germanium cluster, they found a rapid
decrease in the ionization potentials (IPs) for Gen between n = 15
⇑ Corresponding author. Tel.: +86 2884078267; fax: +86 28 85415508.

E-mail address: (S. Shi).
/>2210-271X/Ó 2014 Elsevier B.V. All rights reserved.

and 26, which was very similar to that for silicon clusters [4].
The low-lying stages of Ge2 and GeÀ
2 have also been probed using
negative ion zero electron kinetic energy spectroscopy [7]. Because
of the lack of experimental method to characterize the structure of
germanium clusters, most of the geometrical data come from theoretical calculations. Geometrical and electronic properties of Gen
(n = 5–10) neutrals, cations, and anions have been investigated
using the density functional method of Becke’s three-parameter
hybrid functional with the Perdew/Wang 91 expression by Li
et al. [12]. Yoo and Zeng performed a constrained search for the
geometries of low-lying neutral germanium clusters in the size
range of 21 6 n 6 29 [14]. Wang et al. calculated dipole polarizabilities of Gen clusters at FF level of density functional theory, which
show the dipole moment and polarizabilities of Gen clusters are
sensitively dependent on the cluster geometries and electronic
structures [15]. King et al. reported the effect of electron count
on cluster geometry of nine and ten atom germanium clusters
using B3LYP level of DFT [20].
The pure germanium clusters are chemically reactive and thus
not suitable as a building block of self assembly materials. By an
appropriate choice of the metal dopant, it is possible to design
metallic as well as semiconducting nanotubes using Gen as


9

S. Shi et al. / Computational and Theoretical Chemistry 1054 (2015) 8–15

building blocks. Doped germanium clusters have been performed
the focus of a few experimental and theoretical studies [22–35],
which exhibit many novel properties such as the sizes selectivity,
the highest occupied molecular orbital-lowest unoccupied molecular orbital gap, different charge transfer direction and the magnetic
property. Tai and Nguyen [22] found the structure and stability of
the Ge12Mx clusters with M = Li, Na, Be, Mg, B, Al, and x from À1 to
+1, they obtained the high thermodynamic stability of the icosahedra arises from a combination of their closed crystal field shells,
spherical aromaticity and electrostatic attraction force. Electronic
properties of germanium–fluorine cluster anions (GenFmÀ; n = 1–11,
m = 1–3) were studied by Negishi et al. using photoelectron spectroscopy with a magnetic-bottle type electron spectrometer, which
showed that the doped F atom in GenFÀ deprives each GenÀ cluster
of the excess electron without any serious rearrangement of the
Gen framework [23]. In addition, the geometries, stability, and
electronic properties of TM-doped germanium clusters (TM = Zn,
Fe, Mn, Si, Ni, W, Cr, Cu, Au) [25–35] have also been systematically
investigated by using different method. The remarkable features of
Zn-doped Gen clusters are distinctly different from other TM-Gen
clusters, indication that the growth pattern of the TM-Gen clusters
depends on the kind of doped TM impurity.
Although many studies have been taken on pure germanium
clusters and doped germanium clusters, to our knowledge, surely
systematic and theoretical investigated on aluminum-doped germanium clusters have not been reported so far. In this work, an

investigation on the structures, stabilities, magnetism, and
electronic properties of the Al-doped germanium clusters were calculated using density functional theory by considering a considerable number of structural isomers. In order to reveal the effect of
the doped Al atom to the germanium clusters, in this paper, we
optimize the geometrical structures of GenAl (n = 1–9) clusters by
employing DFT approach to find the structural and stability, and
combined with pure germanium clusters for comparison by using
identical methods and basis sets.

2. Computational details
The geometry optimizations of the Gen+1 and GenAl (n = 1–9)
clusters with spin configurations considered are performed by
using density functional theory (DFT) with the B3LYP exchange–
correlation potential and 6-311G(d) basis sets. The B3LYP method,
it is based on the Becke three-parameter exchange functional and
the Lee, Yang and Parr correlation functional [36,37]. In order to
test the reliability of our calculations, some test calculations are
carried out on Ge2 and Al2 using B3LYP, B3P86, PBE1PBE, and
B3PW91 method with LANL2DZ, Def2-TZVP, and 6-311G(d) basis
sets. The computed spin multiplicities, bond lengths (Re), vibrational frequencies (xe), and dissociation energies (De) of dimers
(Ge2, and Al2) and available experimental and previous theoretical
data are summarized in Table 1. Comparing with the experimental
data, we can find that the B3LYP method with 6-311G(d) basis sets

Table 1
The computed spin multiplicities, bond lengths (Re), vibrational frequencies (xe), and dissociation energies (De) of dimers (Ge2, and Al2) and available experimental and previous
theoretical data.
Molecule

Method

Ge2

B3LYP

B3P86

PBE1PBE

B3PW91

Spin

Re (Å)

xe (cmÀ1)

De (eV)

LanL2DZ
6-311G(d)
Def2-TZVP
LanL2DZ
6-311G(d)
Def2-TZVP
LanL2DZ
6-311G(d)
Def2-TZVP
LanL2DZ
6-311G(d)
Def2-TZVP

3
3
3
3
3
3
3

3
3
3
3
3
3a
–

2.528
2.413
2.407
2.517
2.388
2.385
2.514
2.386
2.383
2.520
2.393
2.390
2.548b
2.44e

250.1
276.6
279.0
256.4
288.0
290.1
259.4

292.0
294.1
255.8
286.5
288.4
281c
274f
286 + 5g,h

2.34
2.87
2.93
2.52
3.02
3.08
2.40
2.89
2.97
2.42
2.89
2.96
2.34d
2.65f
2.70 ± 0.07g,h

LanL2DZ
6-311G(d)
Def2-TZVP
LanL2DZ
6-311G(d)

Def2-TZVP
LanL2DZ
6-311G(d)
Def2-TZVP
LanL2DZ
6-311G(d)
Def2-TZVP

3
3
3
3
3
3
3
3
3
3
3
3
3i
3k

2.855
2.765
2.753
2.834
2.737
2.728
2.834

2.739
2.730
2.839
2.746
2.736
2.7i
2.7k

235.1
252.9
259.6
246.4
266.1
271.4
250.7
268.5
274.8
245.9
264.0
270.0
241.2i
284.2k

1.19
1.29
1.36
1.36
1.47
1.53
1.37

1.48
1.54
1.33
1.44
1.49
1.33j
1.34l

Theory
Experiment
Al2

B3LYP

B3P86

PBE1PBE

B3PW91

Theory
Experiment
a
b
c
d
e
f
g
h

i
j
k
l

Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.
Ref.

[32].
[33].
[34].
[35].
[6].
[7].
[8].
[9].
[39].
[40].
[41].
[42].

10

S. Shi et al. / Computational and Theoretical Chemistry 1054 (2015) 8–15

is more optimal than others. Therefore, the B3LYP/6-311G(d)
scheme is reliable and accurate enough for describing the systems
involving Ge and Al atoms. The B3LYP method or 6-311G(d) basis
sets was successfully used for pure Gen clusters [12,17] and for
GenLi clusters [22], GenCr clusters [31].
In this paper, the conformations of the pure Gen+1 clusters are
obtained firstly by reference to the configurations in Refs.
[13,15,18,19]. During the course of choosing initial structures of
the GenAl clusters, we have considered possible isomeric structures
by placing the Al atom on each possible site of the Gen+1 clusters as
well as by substituting one Ge atom by the Al atom from Gen+1
cluster. Furthermore, different spin states of GenAl clusters are considered and calculated by using the Gaussian 03W package [38],
the optimized results are obtained that the most stable structures
of the Al-doped germanium clusters. The detailed calculated
results and discussions are followed.

3. Results and discussions
3.1. Lowest-energy structures
Using the computation scheme described in Section 2, we have
explored a number of low-lying isomers and determined the lowest-energy of GenAl clusters up to n = 9. The obtained ground state
geometries and some low-lying metastable isomers are shown in
Fig. 1. For proper comparison we have also shown the ground
state geometries of pure Gen+1 (n = 2–9) clusters. The point group
symmetries (PG), the spin multiplicities, the electronic states,

geometry property, and relative energy DE (relative to lowestenergy structure) of the most stable and low-lying GenAl (n = 1–9)
clusters are summarized in Table 2. For Ge2 dimer, the
dissociation energy and vibrational frequency are obtained as
2.87 eV and 276.6 cmÀ1. Our current results are in satisfactory
agreement with the experimental data (dissociation energy
De = 2.7 eV and x = 286 cmÀ1) [8,9]. The Ge–Ge bond length for
Ge2 dimer is predicted to be 2.413 Å, this is also consistent with
results 2.44 Å of experimental [6]. The electronic state and spin
multiplicity are 3Rg and triplet spin state, respectively, which
agrees well with the results of Deutsch et al. [16] and
Bandyopadhyay and Sen [32]. For the GeAl monomer with C1v
symmetries, the optimized results indicate that the quadruple
spin state is lowest energy. Therefore, the quadruple GeAl monomer with a bond length of 2.491 Å is most stable structure, the
corresponding electronic state is 4R.
For Ge3, the isosceles triangle structure is suggested as the lowest energy structure with an apex angle of 83.8° corresponding to
the 1A0 . Our result is excellent with previous theory [19], in which
the isosceles triangle structure with an apex angle 84.9°. The most
stable structure of Ge2Al cluster is also an isosceles triangle structure (3-a). This configuration presents the low spin state of 1A0 . The
Ge–Ge bond length, the Ge–Al bond length, and the vibrational frequencies of ground 1A0 state of Ge2Al cluster is 2.585 Å, 2.408 Å,
and 167.4 cmÀ1, respectively. The linear C1v (Ge–Ge–Al) configuration is also considered, corresponding to 3-b isomer with 1A00 state,
the linear C1v structure is lower in energy than the linear Cs isomers. The linear Ge–Al–Ge geometry with quartet spin multiplicity
is also found to be stable. However, the configuration corresponds
to very high relative energies of 1.120 eV.
The Ge4 is a rhombus structures with C2V symmetry, the corresponding to bond length is 2.475 Å and electronic state is 1A1. Five
kinds of Ge3Al clusters can be optimized to the minima. When
n = 3, the planar structures (3-a) are proved to be the lowestenergy structures, but three-dimensional (3D) structures (3-b, 3-c,
3-d, and 3-e) are not the most stable structures in our calculated
clusters. The Ge–Ge–Ge bond angle (104.3°) of the 3-a isomer,

generated from substitution of Ge4 3-a0 rhombus by Al, is much

larger than that of the Ge3 cluster. The 3-b isomer is a distorted
Y-type structure, which can be described as one Al atom being
bonded on the apical Ge atom in the lowest energy Ge3 cluster. If
one Al or Ge atom is capped on the lowest energy 2-a0 or 2-a,
the 3-c or 3-d isomer may be formed. 3-e is the highest energy
between planar structures and 3D structures. The most stable
structure (3-a) have CS symmetry and 2A0 electronic state.
The most favorable geometry of Ge5 cluster is a distorted trigonal bipyramid structure with D3H symmetry, corresponding to the
1 0
A state. The lowest energy structure of Ge4Al is 4-a in C1 symmetry with 2A electronic state, which is formed by capping one Al
atom on the top of Ge4 cluster. 4-b and 4-c can be viewed as Al
atom substituted a Ge atom from the apical and middle in the
Ge5 isomer, which are obvious higher in energy than the lowest
energy structure 4-a by 0.548 and 0.748 eV, respectively. One planar isomer (4-d) with C1 symmetry behaves the highest energy
(DE = 1.076 eV) among all isomers of Ge4Al clusters. Therefore,
from n = 4, the 3D structures are more stable than the planar structures (4-a > 4-b > 4-c > 4-d).
The distorted octahedron is obtained for Ge6, which has D3 symmetry, corresponding to the electronic is 3A1. Four structures
obtained for Ge5Al clusters have C1 symmetry. The most stable
(5-a) with 2A electronic state, corresponding to the Ge–Ge bond
length, and the Ge–Al bond length are 2.612 Å and 2.802 Å. The
prism structure (5-b) and the distorted octahedron structure (5-c)
usually are considered the most stable, but in our calculation, their
energies higher than the 5-a, which are 0.441 and 0.603 eV, respectively. The structure of 5-d isomer with C1 symmetry and spin multiplicity (PG = 2), which relation energy is 1.144 eV.
In the case of n = 7, the pure Ge7 adopted the pentagonal bipyramid structure with C1 symmetry, corresponding to the electronic
state and Ge–Ge bond length are 1A and 2.681 Å, respectively.
Although geometries structure of Ge7 cluster is same as Refs.
[19,32], the symmetry is different. Four different isomers are found
for Ge6Al. The most stable structure (6-a) is obtained by one Al
atom substitute one Ge atom from the waist of pure Ge7 cluster
(6-a0), which has C1 symmetry, and its the Ge–Ge bond length,

and the Ge–Al bond length are 2.697 Å and 2.636 Å. The other pentagonal bipyramid structure (6-b) is one Al atom substitute one Ge
atom from the top of pure Ge7 cluster (6-a0). The relation energy of
6-c and 6-d are 0.607 and 0.827 eV, respectively.
When the size of Gen clusters is up to 8, one structure, which is
obtained from the pentagonal bipyramid Ge7 (6-a0), is proven to be
stable structure. The Ge–Ge bond length, and the vibrational frequencies of ground 1A state of Ge8 cluster are 2.681 Å, and
86.5 cmÀ1, respectively. The most stable Ge7Al (7-a) cluster can
be generated from one Al atom substitute one Ge atom on the lowest energy Ge8 cluster. It displays C1 symmetry with 2A electronic
state. The 7-b, 7-c, and 7-d isomers have higher energies compared
with the 7-a structure in its ground state by 0.051, 0.249, and
0.641 eV, respectively.
As for Ge9, the lowest-energy structure is a bicapped pentagonal
bipyramid structure with C1 symmetry, corresponding to the spin
multiplicity is 1. The configuration of Ge9 can be easily understood
as growth on the basis of Ge8. Five kinds of stable structures can be
verified to be the minima in Ge8Al. The most stable structure 8-a
with C1 symmetry, it is seen that the Al atom substitutes one Ge
atom of the Ge9 cluster. Other possible isomers (8-b–8-e) have
energies higher than 8-a by 0.202, 0.260, 0.276, 0.905 eV.
Especially, the 8-e isomer is much higher in energy than 8-a by
0.905 eV. Although both Ge8Al 8-c and 8-d isomers are close in
energy, they have different structures.
The ground-state structure obtained for Ge10 has C1 symmetry
and it can be built from Ge9 wedges. The electronic state and the
Ge–Ge bond length are 1A and 2.728 Å, respectively. For Ge9Al,


11

S. Shi et al. / Computational and Theoretical Chemistry 1054 (2015) 8–15

1-a0

1-a

2-a0

2-a

2-b

2-c

3-a0

3-a

3-b

3-c

3-d

3-e

4-a0

4-a

4-b

4-c

4-d

5-a0

5-a

5-b

5-c

5-d

6-a0

6-a

6-b

6-c

6-d

7-a0

7-a

7-b

7-c

7-d

8-a0

8-a

8-b

8-c

8-d

8-e

9-a0

9-b

9-c

9-a

9-d

Fig. 1. Ground-state configurations and low-lying isomers of GenAl (n = 1–9) clusters and the lowest-energy structures of pure Gen+1 (n = 1–9) clusters. The first GenAl
structure is the lowest-energy one for GenAl (n = 2–9).

the four most stable isomers are listed in Fig. 1 (9-a–9-d), although
from 9-a to 9-d isomers are different in structure, they have same

symmetry (C1) and electronic state (2A). The lowest energy isomer
is in C1 symmetry, a multirhombus prism with one side capped Ge


12

S. Shi et al. / Computational and Theoretical Chemistry 1054 (2015) 8–15

Table 2
The point group symmetries (PG), spin multiplicity, electronic state, bond lengths and vibrational frequencies, and relative energies of Gen+1 and GenAl (1–9) clusters. Rav1 and Rav2
denote the average bond lengths of Ge–Ge and Ge–Al, respectively; Freq denotes the lowest vibrational frequency of the Gen+1 and GenAl equilibrium geometry.
Cluster
Ge2
GeAl
Ge3
Ge2Al

Ge4
Ge3Al

Ge5
Ge4Al

Ge6
Ge5Al

Ge7

Ge6Al

Ge8
Ge7Al

Ge9
Ge8Al

Ge10
Ge9Al

Isomer
1-a0
1-a
2-a0
2-a
2-b
2-c
3-a0
3-a
3-b
3-c
3-d
3-e
4-a0
4-a
4-b
4-c
4-d
5-a0

5-a
5-b
5-c
5-d
6-a0
6-a
6-b
6-c
6-d
7-a0
7-a
7-b
7-c
7-d
8-a0
8-a
8-b
8-c
8-d
8-e
9-a0
9-a
9-b
9-c
9-d

PG
D1h
C1v
Cs

Cs
C1v
C2v
C2v
Cs
C1
C3v
Cs
Cs
D3H
C1
C2v
C3v
C1
D3
C1
C1
C1
C1
C1
C1
C1
C1
C1
C1
C1
CS
C1
C1
C1

C1
C1
C1
C1
C1
C1
C1
C1
C1
C1

Spin

State

Rav1 (Å)

Rav2 (Å)

Freq (cmÀ1)

DE (eV)

3
4
1
2
2
4
1

2
4
6
6
6
1
2
4
4
4
3
2
2
4
2
1
2
2
2
2
1
2
2
2
2
1
2
2
2
2

2
1
2
2
2
2

3

2.403
–
2.842
2.585
2.306
5.006
2.475
2.424
2.505
2.664
2.664
2.564
2.645
2.520
2.603
2.682
2.505
2.655
2.612
2.590
2.619

2.611
2.681
2.697
2.693
2.640
2.702
2.681
2.722
2.810
2.671
2.692
2.714
2.736
2.619
2.785
2.706
2.770
2.728
2.729
2.757
2.786
2.737

–
2.491
–
2.408
2.598
2.503
–

2.491
2.573
2.699
2.700
2.817
–
2.919
2.752
2.695
2.594
–
2.802
2.554
2.711
2.572
–
2.636
2.722
2.711
2.690
–
3.323
3.247
2.987
2.581
–
2.625
2.697
2.973
2.998

2.778
–
2.687
2.583
2.590
2.601

276.6
293.0
94.7
167.4
337.7
49.6
53.7
15.3
7.8
135.4
135.4
90.2
38.5
81.2
72.9
54.3
14.6
58.2
20.2
33.3
16.5
24.4
86.5

78.2
75.4
9.5
30.7
86.5
27.8
38.3
42.0
54.5
48.3
46.4
45.7
42.5
34.7
28.1
6.2
34.2
36.9
45.8
31.5

0.00
0.00
0.00
0.00
0.143
1.120
0.00
0.00
1.492

1.624
1.625
2.380
0.00
0.00
0.548
0.748
1.076
0.00
0.00
0.441
0.603
1.144
0.00
0.00
0.040
0.607
0.827
0.00
0.00
0.051
0.249
0.641
0.00
0.00
0.202
0.260
0.276
0.905
0.00

0.00
0.276
0.361
1.043

Rg
4
R
1

A0
A0
1 00
A
4
B2
1
A1
2 0
A
4
A
6
A1
6 0
A
6 0
A
1 0
A

2
A
4
A2
4
A1
4
A
3
A1
2
A
2
A
4
A
2
A
1
A
2
A
2
A
2
A
2
A
1
A

2
A
2 0
A
2
A
2
A
1
A
2
A
2
A
2
A
2
A
2
A
1
A
2
A
2
A
2
A
2
A

1

atom and the other side capped Al atom, which can be view as one
Ge atom capped on the 8-d cluster. Other possible isomers (8-b–8d) have energies higher than 8-a by 0.276, 0.361, and 1.043 eV as
shown in Table 2.
3.2. Relative stability of different sized GenAl clusters
The understanding of the relative stability of different sized
GenAl (1–9) clusters is important for novel cluster-assembled
optoelectronic materials and can provide a good way to show the
relative local stability of small clusters. So the relative stability of
different GenAl clusters can be represented with the average binding energies (Eb), second-order differences of total energies (D2E)
and fragmentation energies (Ef). Firstly, we consider the corresponding Eb, D2E, and Ef of Gen+1 (n = 1–9) clusters to provide an
interpretation. They are expressed as

Eb ½Genþ1 Š ¼ ððn þ 1ÞE½GeŠ À E½Genþ1 ŠÞ=ðn þ 1Þ

ð1Þ

D2 E½Genþ1 Š ¼ E½Genþ2 Š þ E½Gen Š À 2E½Genþ1 Š
Ef ½Genþ1 Š ¼ E½Gen Š þ E½GeŠ À E½Genþ1 Š

ð2Þ
ð3Þ

where E[Ge], E[Gen], E[Gen+1], and E[Gen+2]denote the total energies
of the lowest energy Ge, Gen, Gen+1, and Gen+2 clusters, respectively.
For GenAl (n = 1–9) clusters, the average binding energies (Eb),

second-order differences of total energies (D2E) and fragmentation
energies (Ef) can be calculated by following formulas:

Eb ½Gen AlŠ ¼ ðnE½GeŠ þ E½AlŠ À E½Gen AlŠÞ=ðn þ 1Þ

ð4Þ

D2 E½Gen AlŠ ¼ E½Genþ1 AlŠ þ E½GenÀ1 AlŠ À 2E½Gen AlŠ
Ef ðGen AlÞ ¼ EðGenÀ1 AlÞ þ EðGeÞ À EðGen AlÞ

ð6Þ

ð5Þ

where E[Ge], E[Al], E[GenÀ1Al], E[GenAl], and E[Gen+1Al], respectively, are the total energies of the stable atoms or clusters for Ge,
Al, GenÀ1Al, GenAl, and Gen+1Al.
Based on the above formulas, the calculated results of average
binding energies, second-order differences of total energies and
fragmentation energies are shown in Figs. 2–4. From Fig. 2, in general, it can be seen that binding energies of Gen+1 clusters increase
with cluster size up to n = 9 and contain one minor bump at n = 6
implying that the cluster for n = 6 is more stable than their neighbors, our results are excellent with previous theory [19,33]. When
Al is doped on the pure germanium clusters, the averaged binding
energy increase smoothly with the size of GenAl clusters increases
from 1 to 9, the tendency is almost consistent with the binding
energy of Gen+1 cluster. Although, the average binding energy
increase unceasingly, the average binding energy’s increment
speed slows down gradually for n = 1–3, 3–5, 5–7, 7–9. In general,
the average binding energy grows gradually as clusters size n and


13

S. Shi et al. / Computational and Theoretical Chemistry 1054 (2015) 8–15

Binding Energy per atom (eV)

3.0
2.8
2.6
2.4
2.2

Gen+1
GenAl

2.0
1.8
1.6
1.4
1.2
1.0
0

2

4

6

8

10

Cluster size n
Fig. 2. Calculated binding energy per atom of germanium clusters and Al-doped Ge
clusters (n = 1–9) plotted as function of number of Ge atoms.

The Second-order Differences (eV)

3

Gen+1

2

GenAl

3.3. Homo–Lumo gap

1

0

-1

-2
0

1

2

3

4

5

6

7

8

9

Cluster size z
Fig. 3. Calculated second difference in energies for the germanium clusters and
Al-doped Ge clusters as a function of number of Ge atoms.

4.5

Gen+1

3.5

Gen+1
GenAl

4.0

The calculations of molecular orbital were based on the Hückel

method proposed by Erich Hückel in 1930, which is a linear combination of atomic orbital (LCAO) method. If the closed electronic
configuration of a cluster has a large highest occupied molecular
orbital (HOMO) and lowest unoccupied molecular orbital (LUMO)
gaps, which show the cluster contain high chemical stability. As
seen from Fig. 5, the Ge2 and Ge6 clusters have a larger HOMO–
LUMO gap while the Ge5 and Ge8 clusters have a smaller HOMO–
LUMO gap. However, when Al is doped on the pure germanium
clusters, the GeAl, Ge4Al and Ge8Al clusters have a larger HOMO–
LUMO gap while the Ge5Al, Ge7Al and Ge9Al clusters have a smaller
HOMO–LUMO gap. As shown in Fig. 5, for GeAl, Ge4Al and Ge7Al
clusters, the HOMO–LUMO gaps of GenAl clusters are usually larger
than those of Gen+1 clusters, while for the other clusters, the
HOMO–LUMO gaps of GenAl clusters are usually smaller than those

Gen Al

HOMO-LUMO gap (eV)

Fragmentation Energies (eV)

present local maximum value at n = 1, 3, 5 and 7, implying that the
clusters are more stable than their neighbors. However, as shown
in Fig. 2, the averaged binding energy of GenAl cluster is lower than
that of the Gen+1 cluster. Fig. 3 shows the second-order difference
in cluster total energies, as a function of the cluster size. The
Gen+1 clusters stabilities exhibit pronounced odd–even alternations, but the phenomenon is changed when n = 4, so maxima
are found at n = 1, 3, 6, 8, indicating these clusters possess
relatively higher stability. For GenAl cluster, it is found that the second-order difference in cluster total energies exhibits odd–even
oscillations from 1 to 9. Odd n gives high stability while even n
gives low stability, but the stability of Ge6Al, Ge7Al and Ge8Al

cluster are inverse. A distinct characteristic for the stability of the
GenAl clusters can also be observed, around n = 6, this behavior is
exceptional. As shown in Fig. 4, it is clearly found that the doping
impurity Al atom makes the thermodynamic stability pattern of
the host germanium cluster same from n = 1 to n = 9, except for
n = 6. And the Gen+1 cluster with higher fragmentation energy than
the GenAl clusters, suggesting Gen+1 cluster have higher stability,
except for Ge4, Ge5, and Ge7 clusters. The local maxima of the fragmentation energy of Gen+1 and GenAl clusters appear at 3, 6, 9 and
3, 5, 9. That is to say the Ge4, Ge7, Ge10 and Ge3Al, Ge5Al, Ge9Al
clusters have higher fragmentation energy, indicating that these
clusters are more stable than their neighboring ones.

3.5
3.0
2.5
2.0

3.0

2.5

2.0

1.5

1.0

1.5
0

2

4

6

8

10

Cluster size n
Fig. 4. Calculated fragmentation energies for the germanium clusters and Al-doped
Ge clusters as a function of number of Ge atoms.

0

2

4

6

8

10

Cluster size n
Fig. 5. Calculated HOMO–LUMO gaps for the germanium clusters and Al-doped Ge
clusters as a function of number of Ge atoms.

14

S. Shi et al. / Computational and Theoretical Chemistry 1054 (2015) 8–15

of Gen+1 clusters. From Fig. 5, maxima of GenAl cluster is found at
n = 1, indication GeAl cluster possesses the highest stability.

1.6
1.4

Gen+1

3.4. Electronic properties

1.2

Gen Al

In cluster science, electron affinity (EA) and ionization potential
(IP) are used as important properties to study the change in electronic structure of the cluster size. Vertical electron affinity (VEA)
is defined as the energy difference between the anionic and neutral
clusters with both at the optimized geometry of the anionic cluster,
while the vertical ionization potential (VIP) is defined as the energy
difference between the cationic and neutral clusters with both at
the optimized geometry of the neutral cluster. Chemical hardness
(g) is expressed as g = VIP–VEA based on the basis of a finitedifference approximation and the Koopmans theorem [43], and is
established as an electronic quantity which may be applied in
characterized the relative stability of molecules and aggregate
through the principle of maximum hardness (PMH) proposed by

Pearson [44]. The VIP, VEA and g of the most stable Gen+1 and GenAl
(n = 1–9) clusters are calculated and listed in Table 3. The VIP
shows an oscillating behavior from Ge2 to Ge10, except for Ge9,
the EVA shows an increase from Ge2 to Ge6 and an oscillating
behavior from Ge6 to Ge10, therefore, the Ge3 has the maximum
hardness while Ge6 has the minimum hardness. When Al atom is
doped Gen+1 cluster, it clearly sees that doping with Al atom
reduces the vertical ionization potential and chemical hardness
of germanium clusters, but doped Al atom raise the vertical electron affinity of germanium clusters. For GenAl clusters, The VIP
shows a decrease from Ge2Al to Ge9Al, except for Ge8Al, while
VEA increase with n. In addition, the chemical hardness decreases
with cluster size.

1.0

3.5. Magnetisms
Finally, we comment on the magnetic properties of the Gen+1
and GenAl clusters. In Fig. 6, we compare the magnetic moments
of the Gen+1 clusters with computed values for size of the GenAl
clusters. It is interesting that the atomic averaged magnetic
moments are clearly different between the Gen+1 clusters and the
GenAl clusters, indicating that the magnetic moments for the GenAl
clusters are relatively more larger than the Gen+1 clusters, except
Ge6 cluster. In Gen+1 system, it should be mentioned that the most
stable Ge2 and Ge6 clusters exhibit magnetic moments, but Ge3,
Ge4, Ge5, Ge7, Ge8, Ge9, and Ge10 clusters exhibit nonmagnetic
ground state, it means that 1-a0 and 5-a0 isomers are the magnetic
structures, the other isomers are nonmagnetic structures.
However, when Al atom is doped germanium clusters, the magnetic moment changes discontinuously with the cluster size and
the magnetic moments are decreased with increasing cluster size.

For the GeAl dimmer, it is 1.50 lb, it decreases up to 0.1 lb for n = 9.

Table 3
Vertical ionization potential, vertical electron affinity, chemical hardness of the most
stable Gen+1 and GenAl (n = 1–9) clusters (eV).
Cluster size

n=1
n=2
n=3
n=4
n=5
n=6
n=7
n=8
n=9

Gen+1

GenAl

VIP

VEA

g

VIP

VEA

g

7.627
7.967
7.819
7.956
6.888
7.797
6.875
7.131
7.434

1.751
1.702
1.860
2.125
2.615
1.764
1.956
1.482
1.644

5.876
6.265
5.959
5.831
4.273
6.033
4.919

5.649
5.790

7.599
7.384
7.295
7.444
7.270
7.120
6.239
7.224
7.085

1.139
2.033
1.898
2.531
2.265
2.204
2.108
2.876
3.014

6.460
5.351
5.395
4.913
5.005
4.916
4.131

4.348
4.071

0.8
0.6
0.4
0.2
0.0
0

2

4

6

8

10

Cluster size n
Fig. 6. Size dependence of the average atomic magnetic moments of the lowestenergy Gen+1 and GenAl clusters.

There is sharply decrease from n = 1 (1.50 lb/atom) to n = 2
(0.33 lb/atom) in the curve of magnetic moments, staring from
n = 2, the moments gradually decrease as the cluster sizes increase.
4. Conclusions
In conclusion, we report a systematic study of the geometric
structures, relative stabilities, electronic properties and magnetic
properties of Gen+1 and GenAl (n = 1–9) clusters using density functional theory under the generalized gradient approximation

scheme. Extensive structures and different possible spin states
are carefully investigated. In order to show the properties of Al
atom doped germanium clusters, we also calculate the properties
of pure Gen+1 clusters. The results can be summarized as follows:
(1) According to the optimized equilibrium geometries of the
Gen+1 and GenAl clusters, the growth pattern of the Gen+1
and GenAl clusters is investigated. Theoretical results indicate that the low-lying isomers for the Gen+1 and GenAl clusters become three dimensional structures when the size
n = 4. On the whole, the adopted lowest energy structures
of the GenAl are similar to lowest energy structures of the
Gen+1 clusters.
(2) The stability analysis in relation to the calculation of the
averaged atomic binding energy, the fragmentation energy,
and the second order difference of energy shows that Gen+1
cluster possess relatively higher stability than GenAl cluster.
The average binding energies of the most stable Gen+1 clusters are higher than those of the GenAl clusters. According
to the fragmentation energy and the second order difference
of energy analysis, it is concluded that the small Ge7 and
Ge5Al isomers are the most stable geometries for Gen+1 and
GenAl clusters, respectively.
(3) The HOMO–LUMO gaps are extensively analyzed for Gen+1
and GenAl clusters. The obtained results reveal that the GeAl,
Ge4Al and Ge8Al clusters have a larger HOMO–LUMO gap
while the Ge5Al, Ge7Al and Ge8Al clusters have a smaller
HOMO–LUMO gap. The VIP of the Gen+1 clusters show an
oscillating behavior from Ge2 to Ge10, except for Ge9, the
EVA of the Gen+1 clusters show an increase from Ge2 to Ge6
and an oscillating behavior from Ge6 to Ge10. The VIP of
the GenAl clusters shows a decrease from Ge2Al to Ge9Al,
except for Ge8Al, while VEA of the GenAl clusters increase
with n.

S. Shi et al. / Computational and Theoretical Chemistry 1054 (2015) 8–15

(4) The investigated magnetic moments of the GenAl cluster
indicate that the atomic averaged magnetic moments
decrease with cluster size increasing. Moreover, for the
Gen+1 clusters, the Ge2 and Ge6 clusters exhibit magnetic
moments, but Ge3, Ge4, Ge5, Ge7, Ge8, Ge9, and Ge10 clusters
exhibit nonmagnetic ground state.

Acknowledgements
This research is supported by Cultivating programme of excellent innovation team of Chengdu university of technology (Grant
No. JXTD20130) and Cultivating Programme of Middle-aged
backbone teachers of Chengdu University of Technology. We
acknowledge Project supported by the Scientific Research
Foundation of the Education Department of Sichuan Province,
China (Grant No. 11ZB036 and 11ZB266).
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