JChemPhys 123 244303 cu doped ge cluster

A computational investigation of copper-doped germanium and germanium
clusters by the density-functional theory
Jin Wang and Ju-Guang Han
Citation: J. Chem. Phys. 123, 244303 (2005); doi: 10.1063/1.2148949
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THE JOURNAL OF CHEMICAL PHYSICS 123, 244303 ͑2005͒

A computational investigation of copper-doped germanium
and germanium clusters by the density-functional theory
Jin Wanga͒,b͒
Department of Chemistry, University of Guelph, Guelph N1G 2W1, Ontario, Canada

Ju-Guang Hanb͒,c͒
Department of Chemistry, Jackson State University, Jackson, Mississippi 39217

͑Received 4 October 2005; accepted 10 November 2005; published online 23 December 2005͒
The geometries, stabilities, and electronic properties of Gen and CuGen ͑n = 2 – 13͒ clusters have
been systematically investigated by using density-functional approach. According to optimized
CuGen geometries, growth patterns of Cu-capped Gen or Cu-substituted Gen+1 clusters for the smallor middle-sized CuGen clusters as well as growth patterns of Cu-concaved Gen or Ge-capped
CuGen−1 clusters for the large-sized CuGen clusters are apparently dominant. The average atomic
binding energies and fragmentation energies are calculated and discussed; particularly, the relative
stabilities of CuGe10 and Ge10 are the strongest among all different sized CuGen and Gen clusters,
respectively. These findings are in good agreement with the available experimental results on
−
and Ge10 clusters. Consequently, unlike some transition metal ͑TM͒Si12, the hexagonal
CoGe10

prism CuGe12 is only low-lying structure; however, the basketlike structure is located as the
lowest-energy structure. Different from some TM-doped silicon clusters, charge always transfers
from copper to germanium atoms in all different sized clusters. Furthermore, the calculated highest
occupied molecular orbital and lowest unoccupied molecular orbital ͑HOMO-LUMO͒ gaps are
obviously decreased when Cu is doped into the Gen clusters, together with the decrease of
HOMO-LUMO gaps, as the size of clusters increases. Additionally, the contribution of the doped Cu
atom to bond properties and polarizabilities of the Gen clusters is also discussed. © 2005 American
Institute of Physics. ͓DOI: 10.1063/1.2148949͔
I. INTRODUCTION

In recent years, clusters of group-14 elements have attracted attention because they are important for fine processing of semiconductors and synthesizing novel materials. Previous investigations indicate that the pure silicon is
unfavorable for forming large-sized clusters and bulk solids;
however, the encapsulation of transition metal in the largesized silicon cluster contributes to enhancing stability of pure
silicon clusters and simultaneously exhibiting many magic
behaviors, e.g., sized selectivities, different frontier orbital
properties, and polarizabilities.1–15 Compared with silicon
and transition-metal-͑TM͒ doped silicon clusters, present investigations on the germanium clusters mainly focus on different sized pure germanium clusters or halogen-doped germanium clusters.16–22 For different sized Gen clusters, many
properties such as geometries, binding energies, ionization
potentials, electron affinities, dipole polarizabilities, etc., are
calculated by using different theoretical methods. On the
other hand, atomization enthalpies and enthalpies of formation for the different sized germanium clusters are measured
through Knudson effusion mass spectrometry.23 The eleca͒

Electronic mail:
On leave from National Synchrotron Radiation Laboratory, University of
Science and Technology of China, Hefei 230026, People’s Republic of
China.
c͒
Author to whom correspondence should be addressed. FAX: 86-5515141078,. Electronic mail:
b͒

0021-9606/2005/123͑24͒/244303/12/$22.50

tronic binding energies of the Gen ͑n = 4 – 32͒ clusters and
halogen-doped Gen ͑n = 4 – 20͒ clusters had been measured
by aid of photoelectron spectroscopy experiment.24 In addition, the photoionization thresholds of the Gen ͑n = 2 – 57͒
and Snn ͑n = 2 – 41͒ clusters were examined by laser photoionization coupled with reflectron time-of-flight mass
spectrometry;25 their experimental results indicated that there
was a major maximum of ionization potential ͑IP͒ for the
Gen clusters at n = 10 and a rapid decrease of IP at 15ഛ n
Ͻ 26. Although the Gen clusters were studied in detail
through the theoretical and experimental methods, a few theoretical investigations on the transition-metal-doped germanium clusters are reported. Ab initio pseudopotential planewave method was employed to investigate the encapsulated
caged TMGen ͑n = 14– 16͒ clusters.26 Their theoretical results
revealed that growth behavior of metal-encapsulated germanium clusters was different from that of metal-encapsulated
silicon clusters and had large highest occupied molecular
orbital-lowest unoccupied molecular orbital ͑HOMOLUMO͒ gaps, etc. Additionally, they also adopted identical
calculation method to study structures of ZnGe12 and
CdSn12,27 and pointed out that these clusters had perfect
icosahedral symmetry and large HOMO-LUMO gap of about
2 eV.
So far, there is no theoretical investigation on the Cudoped germanium clusters reported. In this work, a detailed
investigation on the structures, stabilities, and bonding properties of the Cu-doped germanium clusters are calculated at

123, 244303-1

© 2005 American Institute of Physics

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244303-2

J. Wang and J.-G. Han

UB3LYP/LanL2DZ level. In order to examine the effect of
the doped Cu atom to the germanium clusters, geometry optimizations of the pure germanium clusters are also calculated by using identical methods and basis sets.
II. COMPUTATIONAL DETAILS

The geometry optimizations of the Gen and CuGen ͑n
= 2 – 13͒ clusters with spin configurations considered are performed by using density-functional theory ͑DFT͒ with the
unrestricted B3LYP exchange-correlation potential28,29 and
an effective core potential LanL2DZ basis sets. The standard
LanL2DZ basis sets,30 which provide an effective way to
reduce difficulties in calculations of two-electron integrals
caused by heavy transition-metal atom, are employed. In previous paper, the LanL2DZ basis sets of effective core potential ͑ECP͒ theory are proven to be reliable for the geometries,
stabilities, and electronic properties of the TM-Sin and Gen
systems.31–35,21 In order to test the reliability of our calculation, the Cu2 dimer is calculated. The theoretical results indicate that the singlet Cu2 dimer is the most stable state, and
that the Cu–Cu bond length ͑2.259 Å͒ and vibrational frequency ͑256 cm−1͒ obtained by using LanL2DZ basis sets
are in good agreement with the experimental values of
2.22 Å and 266 cm−1, respectively.36,37
All theoretical calculations are performed with
38
GAUSSIAN-03 program package. In this paper, equilibrium
geometries of the Gen clusters are optimized first. On the
basis of the optimized Gen geometries, different evolution
patterns for determining the different sized CuGen isomers,
i.e., Cu-capped, Cu-substituted, and Cu-concaved patterns as
well as Ge-capped pattern, are taken into accounts.
In order to test the basis sets, the stabilities of CuGen
clusters are calculated and discussed first. The basis sets labeled GEN, which are the combinations of LanL2DZ and
6-311+ G͑2d͒ basis sets, are employed for the Cu and Ge

atoms, respectively. The calculated results by using the GEN
basis sets indicate that the total energy of every isomer decreased; however, the energetic ordering of the competitive
isomers for a definite-sized CuGen clusters is essentially unchanged as compared to the results calculated by using the
LanL2DZ basis sets. Consequently, the LanL2DZ basis sets
are reliable and accurate enough to be applied to describe the
properties of the CuGen and Gen clusters in this paper. Furthermore, all the calculated results in this paper are calculated at the UB3LYP/LanL2DZ level. On the basis of the
calculated structures with different spin states considered, the
optimized results show that the most stable structures of the
Cu-doped germanium clusters are spin doublet states in all
isomers. However, in the case of the pure germanium clusters, the most stable state of Ge2 dimer is triplet spin state
and the most stable state of other different sized germanium
clusters corresponds to singlet spin state in all isomers.
III. RESULTS AND DISCUSSIONS
A. Growth behavior of different sized germanium
clusters

Different spin states of the Ge2 dimer are considered,
and the optimized results indicate that the triplet spin state is

J. Chem. Phys. 123, 244303 ͑2005͒

lower in total energy than the singlet spin state by 0.643 eV.
Therefore, the 3⌺−g state is found to be ground state for the
Ge2 dimer. This result is in agreement with previous calculation by using different density-functional methods.18,39 The
open triangular structure, the closed triangular structure, and
the linear structure of the Ge3 isomer are considered; the
most stable geometry corresponds to an open triangular
structure with spin singlet state. In addition, the closed triangular structure with spin triplet state can be also optimized to
be a local minimum; however, its total energy is higher by
0.259 eV than that of the open triangular structure with singlet spin state. According to previous calculation on the Ge3,

the electronic state of 1A1 is assigned as the lowest-energy
state which is in agreement with our calculated results.18,39
As shown in Fig. 1, two stable Ge4 structures, i.e., a planar
rhombus 4a and a pyramid 4b, can be generated by capping
Ge atom on the surface site of the stable Ge3 frame and
optimized to be a local minimum. Furthermore, the planar
rhombus 4a structure is obviously more stable than the pyramid 4b structure because the total energy of the former is
distinctly lower than that of the latter by 2.122 eV. Therefore, it is confirmed that the lowest-energy Ge4 cluster is a
planar rhombus structure with electronic state of 1A1 which
is similar to the lowest-energy Si4 isomer and the previous
calculations.18,39
Two stable 5a and 5b Ge5 structures can be described as
the distorted trigonal bipyramid and nonplanar pentagon,
which are generated by capping one Ge on the Ge4 4a. As
seen from the total energy, the distorted trigonal bipyramid
5a isomer is more stable than the nonplanar pentagon 5b
isomer. As seen from our optimized geometries, the lowestenergy Ge5 isomer is a distorted trigonal bipyramid D3h
structure having 1A1Ј character which supports the previous
calculated results by using different density-functional
methods.39 Additionally, the previous ab initio pseudopotential configuration-interaction ͑CI͒ calculation suggests that
the triplet spin state of the standard trigonal bipyramid with
D3h symmetry is a ground state.40 However, our calculations
do not support this earlier calculation because this structure
is optimized to be an unstable structure with two imaginary
frequencies. In analogy to the Ge5 isomer, the out-of-plane
edge-capped patterns can also produce the stable Ge6 structures: one is a bicapped rhombic pyramid 6a and the other is
a boatlike 6b structure. A stable Ge6 6a isomer is described
as one Ge atom being out-of-plane edge-capped between two
Ge atoms of bent rhombus. As seen from Table I, the total
energy of the bicapped rhombic pyramid structure is lower

than that of the boatlike structure by 0.783 eV, indicating
that the stability of the bicapped rhombic pyramid is stronger
as compared to that of the boatlike structure. Furthermore,
the electronic state of the lowest-energy Ge6 6a cluster is
shown to be 1A in our calculation, which is slightly different
from the previous report, 1A1, of the identical structure with
C2v structure;18,39 however, its total energy of the bicapped
rhombic pyramid C2v isomer ͑−22.625 642 3 hartree͒ is almost the same as that of the identical structure with C1 symmetry ͑−22.625 643 7 hartree͒ in this work.
According to vibrational frequency analysis, three Ge7
structures are proven to be the stable structures. One is a

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244303-3

Copper-doped germanium clusters

J. Chem. Phys. 123, 244303 ͑2005͒

FIG. 1. All the equilibrium Gen ͑n = 2 – 13͒ clusters. The stars show the lowest-energy Gen ͑n = 2 – 13͒ structures.

pentagonal bipyramid Ge7 7a cluster, which can be produced
by two Ge atoms being symmetrically face capped on the top
of the bent rhombus of the Ge5 5a cluster. The other belongs
to a typical multirhombus 7b structure, which is composed of
the planar or bent rhombus. A stable 7c structure is generated
with one Ge atom being face capped on the top of the boatlike Ge6 6b cluster. As illustrated from Table I, the pentagonal bipyramid, which is the lowest-energy geometry as compared to the multirhombus structure, is still a dominant
structure. Therefore, our calculated results are in good agreement with the previous full-potential linear muffin-tin orbital
molecular-dynamics ͑FP-LMTOMD͒ method calculations.41
Furthermore, one pentagonal bipyramid D5h isomer with

electronic state of 1A1Ј is the lowest-energy structure. Similar
to the Ge7 cluster, the pentagonal bipyramid Si7 cluster was
also proven to be ground state in the previous calculation.42
As far as the Ge8 cluster is concerned, one structure,
which is obtained from the pentagonal bipyramid Ge7 7a, is
proven to be stable structure. Another face-capped pentagonal pyramid or multirhombus ͑1Ge–2Ge–3Ge–4Ge rhombus
and 5Ge–6Ge–7Ge–8Ge rhombus͒ 8b cluster is verified to
be an equilibrium structure; however, its stability is weaker
than the 8a due to its total energy that is slightly higher than

that of the 8a isomer. Previous FP-LMTO calculation pointed
out that the identical 8b cluster with Cs symmetry was the
most stable structure.41 However, our calculation proves that
the identical structure is not the most stable structure because
its total energy is slightly higher than that of the 8a cluster by
0.09 eV. On the basis of the Ge7 7c isomer, one stable 8c
structure is formed with respect to one Ge atom being the
face capped on the bottom of the Ge7 7c isomer. As illustrated in Table I, the total energy of the face-capped hexagonal bipyramid is higher than those of other isomers. Hence,
the lowest-energy Ge8 8a isomer can be described as having
1
A character.
When the size of Gen clusters is up to 9, four kinds of
structures can be verified to be the minima. The lowestenergy 9a and low-lying 9c and 9d isomers are described as
one Ge being face capped on the top, bottom, and side of the
multirhombus Ge8 isomer, respectively. A stable 9b structure
is generated by capping two Ge atoms on the surface sites of
the Ge7 7a isomer. As compared with the low-lying Ge9
9cstructure, it should be mentioned that the analogous structure of the Si9 isomer is the most stable structure.42
In this work, four kinds of Ge10 clusters are found as
minima. The lowest-energy 10a structure with electronic

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244303-4

J. Chem. Phys. 123, 244303 ͑2005͒

J. Wang and J.-G. Han

TABLE I. Geometries, and total energies of CuGen and Gen ͑n = 2 – 13͒ clusters. Sym means point-group symmetry. State denotes electronic state; RCu–Ge and
RGe–Ge represent the shortest Cu–Ge and Ge–Ge bond lengths, respectively. ET denotes the total energies of different CuGen and Gen isomers. ⌬E means
relative energies of every isomer compared to that of the lowest-energy isomer for identical-sized cluster.
RGe–Ge
͑Å͒

ET
͑hartree͒

2.524

2.488

−203.665 109 9

⌬E
͑eV͒

2.566
2.429

2.604
2.458

−211.228 159
−211.227 946 1

C1͑aЈ͒
C2v͑bЈ͒
Cs͑cЈ͒

2

A
A2
2
AЈ

2.617
2.423
2.497

2.58
2.528
2.523

−215.023 147 2
−215.007 973 5
−215.031 680 9

Cs͑aЈ͒

C1͑bЈ͒
C1͑cЈ͒
C1͑dЈ͒
C1͑eЈ͒

2

AЉ
A
2
A
2
A
2
A

2.489
2.578
2.531
2.593
2.495

2.682
2.579
2.687
2.719
2.506

−218.813 250 7
−218.812 255 3

−218.796 507 2
−218.806 640 2
−218.795 554 7

2

A
AЉ
2
A
2
A

2.555
2.576
2.509
2.497

2.549
2.578
2.574
2.537

−222.578 322 6
−222.588 611 7
−222.595 596 7
−222.595 796 4

0.475
0.195

0.005

Ge7

2

0.115
0.3
0.351
0.147

Ge8

CuGe8

CuGe9

CuGe10

CuGe11

CuGe12

CuGe13

0.006
0.232
0.645

0.027

0.456
0.179
0.482

2.549
2.415
2.424
2.414
2.469

2.577
2.699
2.541
2.567
2.642

−226.371 077 4
−226.364 274 1
−226.362 396 7
−226.369 904 3
−226.375 306 3

C1͑aЈ͒
C1͑bЈ͒
C1͑cЈ͒

2

A
A

2
A

2.404
2.504
2.547

2.647
2.627
2.575

−230.177 727 9
−230.160 815 5
−230.142 760 2

0.46
0.951

D4d͑aЈ͒
C1͑bЈ͒
C1͑cЈ͒
C1͑dЈ͒

2

A1
A
2
A
2

A

2.551
2.626
2.499
2.474

2.786
2.59
2.587
2.581

−234.001 926
−233.977 704 2
−233.922 324 7
−233.949 378 3

0.659
2.166
1.43

C1͑aЈ͒
C1͑bЈ͒
C2v͑cЈ͒
C1͑dЈ͒

2

A
A

2
A2
2
A

2.543
2.554
2.468
2.508

2.619
2.621
2.627
2.552

−237.765 713 4
−237.764 634 4
−237.762 001 5
−237.746 072

0.029
0.101
0.534

C1͑aЈ͒
C1͑bЈ͒
Cs͑cЈ͒
C1͑dЈ͒
C1͑eЈ͒
C1͑fЈ͒

2

A
A
2
AЉ
2
A
2
A
2
A

2.643
2.58
2.53
2.55
2.494
2.682

2.682
2.554
2.634
2.556
2.513
2.538

−241.536 991 4
−241.531 088 3

−241.534 156 1
−241.543 187 8
−241.519 539 9
−241.539 065 3

0.169
0.329
0.246

C1͑aЈ͒
C1͑bЈ͒
C1͑cЈ͒
C1͑dЈ͒

2

2.512
2.6
2.692
2.705

2.499
2.5
2.557
2.519

−245.282 702 6
−245.320 642 5
−245.313 211 8
−245.310 804 3

1.032

2

2

2

A
A
2
A
2
A
2

state of 1A1 can be depicted as tetracapped trigonal prism
structure. Another stable-caged 10b isomer can be described
as irregular pentagonal prism structure. Different from the
two equilibrium structures above, the novel basketlike 10c

Ge5

Ge8

A
A
2
AЉ

2
AЈ
2
A

2

AЈ1
A

2.587
2.45

−18.838 419 9
−18.826 697 2

0.319

C1͑a͒
C1͑b͒

1

A
A

2.601
2.514

−22.625 643 7

−22.596 841 5

0.784

D5h͑a͒
C2v͑b͒
Cs͑c͒

1

A1Ј
A1
1
A1

2.789
2.519
2.507

−26.419 340 8
−26.386 127 2
−26.409 663 4

0.904
0.263

1

1

A
AЈ
1
A1

2.533
2.572
2.576

−30.177 205 7
−30.174 003 5
−30.170 533 2

0.087
0.182

0.288

C1͑aЈ͒
C1͑bЈ͒
Cs͑cЈ͒
Cs͑dЈ͒
C1͑eЈ͒

2

2.122

1

A
A

2

−15.057 266 9
−14.979 266

D3h͑a͒
C1͑b͒

C1͑aЈ͒
C1͑bЈ͒

2

C1͑aЈ͒
Cs͑bЈ͒
C1͑cЈ͒
C1͑dЈ͒

2.563
2.843

1

−207.436 551 9
−207.443 949 3
−207.433 372 5

CuGe7

A1
A

C2v͑a͒
C1͑b͒

2.434
2.457
2.577

2

−11.250 610 4

Ge4

2.456
2.421
2.604

CuGe6

2.378

C 2v

B2
A1

2
A

2

A1

Ge3

C2v͑aЈ͒
C2v͑bЈ͒
C1͑CЈ͒

CuGe5

−7.460 758 8

1

CuGe3

2

2.548

3

2

0.201

⌺−g

Dϱh

2

2

ET
͑hartree͒

Ge2

C 2v

B1

RGe–Ge
͑Å͒

Sym

CuGe2

⌬E
͑eV͒

State

Cluster

Sym

CuGe4

State

RCu–Ge
͑Å͒

Cluster

Ge9

Ge10

C1͑a͒
Cs͑b͒
C2v͑c͒

1

1

1

1

Cs͑a͒

Cs͑b͒
C1͑c͒
Cs͑d͒

1

AЈ
AЈ
1
A
1
AЈ

2.6
2.518
2.57
2.489

−33.976 275 2
−33.940 289 9
−33.975 948 5
−33.950 273 5

0.979
0.009
0.708

C3v͑a͒
C1͑b͒
Cs͑c͒

C1͑d͒

1

A1
A
1
AЈ
1
A

2.626
2.553
2.581
2.561

−37.777 426 5
−37.758 821 6
−37.715 344 1
−37.732 405 8

0.506
1.689
1.225

A

2.598

−41.536 602 9

1

1

Ge11

C1

1

Ge12

C1͑a͒
C1͑b͒
C1͑c͒

1

A
A
1
A

2.562
2.58
2.489

−45.307 604 8
−45.299 904 1

−45.298 562 3

C1

1

2.503

−49.062 096 7

1

0.21
0.246

0.643
0.112
Ge13

A

0.202
0.268

and polyhedral 10d structures are born. However, the total
energy of the 10c isomer is obviously higher than those
of the 10a and 10b isomers by 1.688 and 1.183 eV, respectively.

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244303-5

Copper-doped germanium clusters

Compared to other sized Gen clusters, one Ge11 structure
can be found as minimum and this structure can be regarded
as one Ge atom being face capped the top of the tetracapped
trigonal prism Ge10 cluster. When the size of Gen cluster
keeps increasing to n = 12, three different geometries can be
found as minima in present calculation. The basketlike structure being optimized to be an equilibrium structure is more
stable than another cagelike stable structure which is composed of the double hexagonal Ge6 ring. As far as the Ge13
cluster is concerned, one basketlike 13a structure is found as
minimum and this geometry appears floppy.
In conclusion, for the growth pattern of the Gen clusters,
the in-plane-capped Ge atom only appears in the very smallsized clusters and the out-of-plane edge-capped or facecapped Ge atom is dominant in the different sized Gen clusters.

B. Growth behavior of different sized copper-doped
germanium clusters

The possible CuGe2 geometries such as two linear isomers and a triangular structure are considered; only the C2v
CuGe2 structure with doublet spin configuration involved in
the Cu atom directly being capped on the Ge2 or substitution
of the central Ge in the Ge3 cluster is optimized to be the
most stable structure. The identical structure with spin quartet state is also optimized to be the stable structure; however,
its total energy is obviously higher than that of spin doublet
state by 1.089 eV. Therefore, the electronic state of the
lowest-energy CuGe2 isomer can be described as 2B1.
Three kinds of CuGe3 clusters can be optimized to be the
minima at the UB3LYP/Lan2DZ level. Interestingly, two
rhombic structures, which have identical C2v symmetry, can

be found as the different isomers with different Ge–Cu–Ge
angles, indicating that the two rhombic structures are generated from different precursory molecules. The Ge–Ge–Ge
bond angle ͑130.7°͒ of the rhombus 3aЈ isomer, generated
from substitution of Ge4 4a rhombus by Cu, is much larger
than that of the Ge3 cluster. However, the Ge–Ge–Ge bond
angle of another rhombus 3bЈ isomer, described as Cu being
capped on the Ge3 isomer, is very close to that of the Ge3
structure. When one Ge atom in the Ge4 4b is substituted by
Cu, a new pyramid 3cЈ structure is formed. Furthermore, the
rhombus 3bЈ CuGe3 structure is more stable than the pyramid 3cЈ structure in that the total energy of the former is
lower than that of the latter. On the basis of the most stable
CuGe3 3bЈ structure, the spin quartet state is considered and
optimized to be a transition state with one imaginary frequency. Therefore, the 3bЈ isomer is the lowest-energy isomer which has electronic state of 2A1. According to the discussion above, it is concluded that the dominant growth
pattern for the small CuGen isomers is still described as Cu
being capped on the Gen clusters.
Two stable CuGe4 structures are considered. One is obtained from the Cu substitution of the apical Ge atom in the
Ge5 5a isomer. The new stable CuGe4 4bЈ isomer is yielded
with the top Ge atom in the Ge5 5b cluster being substituted
by Cu. The former is more stable than the latter; therefore,
the 4aЈ is the lowest-energy isomer.

J. Chem. Phys. 123, 244303 ͑2005͒

Three kinds of CuGe5 isomers can be generated as the
minima. Similarly, two stable CuGe5 structures can be
viewed as being generated from the Ge6 6a cluster. An analogous bicapped rhombic pyramid 5aЈ structure is formed
when the bottom Ge atom in the Ge6 6a isomer is substituted
by copper. In addition, the boatlike Ge6 structure is distorted
into a planar CuGe5 5bЈ isomer when one Ge atom in the
Ge6 6b isomer is substituted by Cu. However, this stable

structure is much higher in total energy than the 5aЈ isomer.
One notes that when the top Ge atom in the Ge6 6a isomer is
substituted by a copper another different 5cЈ isomer is obtained as compared to the 5aЈ isomer. However, the total
energy of the 5cЈ cluster is obviously lower than that of the
5aЈ cluster by 0.232 eV and it corresponds to the lowestenergy cluster with electronic state of 2AЈ. Therefore, the
different substituted positions of the Cu atom in the Gen
clusters lead to the different energetic CuGen−1 isomers.
As mentioned above, when copper replaces different Ge
atom in the multirhombus Ge7 7b, different energetic CuGe6
͑6aЈ, 6bЈ, and 6cЈ͒ isomers can be produced. Moreover, the
substitution of the surface Ge atom by Cu in the Gen clusters
causes the distortion of geometry. In addition, it should be
pointed out that the most stable copper-doped Gen clusters
may not be generated directly from the most stable Gen clusters and it depends on the substituted pattern or position. For
example, on the basis of the lowest-energy 7a structure, only
the substitution of the bottom Ge atom in the Ge7 7a cluster
can result in the stable 6dЈ CuGe6 structure; however, this
isomer is not the lowest-energy CuGe6 isomer. Besides the
mentioned equilibrium CuGe6 structures, a new stable 6eЈ
structure can be born and is described as Cu directly being
capped on the boatlike Ge6 6b geometry. For the lowestenergy CuGe6 6aЈ structure, its electronic state can be assigned as 2AЉ.
On the basis of the Ge8 8a cluster, when one Ge atom in
bent rhombus is substituted by Cu, one stable 7aЈ structure is
yielded. Furthermore, when different Ge atom in the facecapped pentagonal pyramid or multirhombus Ge8 8b cluster
is substituted by Cu, different energetic 7bЈ and 7cЈ isomers
can be yielded. The calculated total energy reveals that the
replacement of the bottom Ge atom is superior to the top Ge
atom in that the stability of the 7cЈ isomer is stronger than
that of the 7bЈ isomer. Compared with other CuGe7 isomers,
the most stable 7dЈ structure is formed when the top Ge atom

in the face-capped hexagonal bipyramid Ge8 8c isomer is
substituted by Cu. In addition, the spin quartet state of the
CuGe7 7dЈ structure is also considered and optimized to be a
stable structure; furthermore, its total energy of spin quartet
state is higher than that of spin doublet state by 0.977 eV. It
should be mentioned that the electronic state of the lowestenergy doublet CuGe7 cluster is 2A.
Beginning from the CuGe8 cluster, a copper-concaved
structure appears; however, Cu is not totally encapsulated in
the Ge8 cage. It can be seen that the concaved structure can
keep the analogous framework as the original clusters; however, the surface-inserted or extracapped Ge atom on the
small CuGen−1 clusters causes obvious deformation of the
framework of original clusters. Although the concaved
CuGe8 8bЈ structure beings to be formed, it does not belong

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244303-6

J. Wang and J.-G. Han

to the lowest-lying isomer which is reflected from its total
energy. As observed from the optimized structures, the stable
CuGe8 8cЈ, CuGe8 8aЈ, and CuGe8 8dЈ isomers are formed
when the different Ge atoms in Ge9 9a, Ge9 9b, and Ge9 9d
are substituted by Cu, respectively. In addition, when one Ge
atom in the Ge9 9c isomer is replaced by Cu, the most stable
CuGe8 8eЈ isomer can be obtained and optimized to be the
lowest-energy isomer.
Three different stable CuGe9 isomers are located at
UB3LYP/Lan2LDZ level. On the basis of the Ge9 9c cluster,

the different Cu added patterns, i.e., the capped and concaved patterns, can form different energetic CuGe9 9aЈ and
9bЈ isomers, respectively. However, total energy of the 9bЈ
cluster is obviously higher than that of the 9aЈ cluster, indicating that Cu is easily concaved into the Gen clusters at the
beginning of the CuGe9 cluster. Different from the CuGe8
cluster, the dominant CuGe9 structure varies from the
surface-inserted Cu atom to the Cu-concaved structure. As
illustrated in Table I, the total energy of the concaved 9aЈ
structure is lower by 0.95 eV than that of the surface-inserted
9cЈ structure. Therefore, the 9aЈ isomer is selected as the
lowest-energy isomer.
In analogy to the CuGe9, although the surface-inserted
CuGe10 structure is still the stable structure; however, it is no
longer candidate for the lowest-energy state because of its
higher total energy as compared to other isomers. In this
sized clusters, two different encapsulated structures are considered. One is a typical multirhombus-caged D4d 10aЈ structure, and the other 10bЈ cluster being described as the Cuencapsulated face-to-face pentagonal Ge10 cage is generated
from the Cu concaved in the Ge10 10b cluster. The surfacecapped CuGe10 10cЈ cluster is formed with Cu being absorbed on the surface of the basketlike Ge10 10c cluster. Obviously, stability of the 10cЈ isomer is weaker than those of
the other two encapsulated structures due to its relatively
higher total energy on the potential-energy surface. Additionally, when one Ge atom is substituted by Cu in the Ge11
isomer or Cu is inserted in the Ge10 10a isomer, a stable
CuGe10 10dЈ structure is found and its total energy is obviously higher than that of the encapsulated structure. Therefore, the encapsulated 10aЈ isomer is the lowest-energy isomer with electronic state of 2A1.
Although only one stable Ge11 cluster is yielded, the
copper-doped Ge11 cluster can produce more stable isomers
or all the isomers are generated from the small-sized CuGen
clusters by addition of an extra Ge atom. For example, on the
basis of the face-to-face pentagonal CuGe10 10bЈ cluster, the
lowest-energy CuGe11 11aЈ structure can be formed with one
Ge atom being face capped the top of the pentagon Ge5
clusters. In addition, on the basis of the lowest-energetic
multirhombus 10aЈ structure, one stable CuGe11 11bЈ isomer
which has higher energy than 11aЈ is produced when the

11th Ge atom is edge capped on the rhombus Ge4 cluster and
the upper bent rhombus in the CuGe10 10aЈ cluster is destroyed. Analogous to the 11aЈ structure, a face-to-face pentagon 11cЈ isomer is optimized to be stable structure; however, the face-to-face pentagon 11cЈ isomer is staggered as
compared to the parallel face-to-face pentagon 11aЈ cluster.
According to the calculated total energy, the stability of the

J. Chem. Phys. 123, 244303 ͑2005͒

11cЈ isomer is weaker than that of the 11aЈ isomer; a new
stable 11dЈ isomer, which is higher in total energy than the
11aЈ cluster, is yielded when the 11th Ge atom is capped the
surface site of the CuGe10 10bЈ. Hence, it is exhibited that
the unfavorable additional pattern on the most stable smallsized CuGen structures can still yield the relatively highenergetic large-sized isomers.
Guided by the growth pattern of the CuGe11 isomer, six
different CuGe12 isomers are obtained. One equilibrium 12aЈ
structure is formed with one Ge atom being edge capped the
top of the upper hexagon 11aЈ cluster. Additionally, when the
Ge atom is edge capped on the low pentagon in the 11dЈ
cluster, a new equilibrium 12bЈ structure is yielded. The lowlying 12bЈ structure is also thought as Cu being encapsulated
into the hexagonal-prism-caged Ge12 12b structure. Surprisingly, the total energy of the 12bЈ cluster is higher than that
of the 12aЈ and the stability of the 12bЈ isomer is weaker
than that of the 12aЈ cluster. However, for the TM-doped Si12
cluster, the analogous concaved hexagonal prism structure is
usually the most stable structure.4,31,43–45 This finding indicates that the TM-doped Gen clusters have different growth
behavior as compared to the TM-doped Sin clusters. On the
basis of the basketlike Ge12 12a structure, one stable 12cЈ
structure can be formed with Cu being directly encapsulated
in the center site of the Ge12 cage. As shown in Fig. 2, the
Cu-doped Ge12 cage does not distort the Ge12 frame and it
changes the stability of the Ge12 structure. On the basis of the
Ge12 12c structures, the high-energetic Cu-doped CuGe12

12eЈ isomer is formed when Cu is directly encapsulated in
the Ge12 cage. Except for the encapsulated hexagonal prism
12bЈ isomer, another low-lying encapsulated bicapped pentagonal prism 12fЈ cluster is also obtained. Surprisingly, the
most stable 12dЈ CuGe12 structure is not an encapsulated
structure, but is a concaved basketlike structure. Similar to
our results, Kawamura et al. pointed out that the basketlike
TiSi12 isomer has lower energy than that of the cage
isomer.46
On the basis of the pure Ge13 cluster, one equilibrium
CuGe13 13aЈ structure is generated when Cu is directly encapsulated in the Ge13 cage. As observed in Fig. 2, Cu doped
in the center site of the Ge13 cage keeps the analogous framework as the Ge13 cage and does not obviously distort the
geometry of the Ge13 cage. Analogous to CuGe12, the lowestenergy CuGe13 13bЈ isomer is optimized to be an irregular
basketlike structure and it is more stable than the 13aЈ isomer. In addition, two different CuGe13 13cЈ and 13dЈ isomers are generated when one Ge atom is edge capped on the
CuGe12 12fЈ and CuGe12 12bЈ isomers. Furthermore, the
CuGe13 13cЈ and CuGe13 13dЈ clusters are higher in the total
energies than the 13bЈ cluster.
In conclusion, three kinds of growth patterns, i.e., Cucapped or concaved Gen clusters, Cu-substituted Gen+1 clusters, as well as Ge-capped CuGen−1 clusters, are considered
for the different sized CuGen clusters. For the CuGe2 and
CuGe3 clusters, Cu directly capping on the identical-sized
Gen clusters are dominant growth pattern. However, for other
small-sized CuGen ͑n = 4 – 8͒ clusters, Cu directly substituted
Ge atom of the Gen+1 clusters to form the CuGen clusters is

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Copper-doped germanium clusters

J. Chem. Phys. 123, 244303 ͑2005͒

FIG. 2. All the equilibrium CuGen ͑n = 2 – 13͒ clusters. The starts show the lowest-energy CuGen ͑n = 2 – 13͒ structures.

dominant growth pattern. When the size of CuGen exceeds 8,
two dominant different growth patterns for the middle-sized
or large-sized clusters are considered. One is Cu directly
being inserted into the Gen framework and the other is the
Ge edge capped on the CuGen−1 framework. It can be ex-

pected that Ge face-capped pattern possibly becomes the
dominant pattern with the growth of the CuGen clusters
when the size of the CuGen clusters is larger than 13. Additionally, it should be mentioned that Cu-doped Gen structures
contribute to stabilizing Gen clusters.

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J. Wang and J.-G. Han

J. Chem. Phys. 123, 244303 ͑2005͒

changed. As observed from HOMO of the CuGe6 6eЈ cluster,
the bonding of 1Ge–6Ge and 3Ge–4Ge appears ␲-type bond.
In addition, the delocalized d-p ␲-type bond is formed between Cu and 1Ge–3Ge–4Ge–6Ge, and it indicates that the
capped Cu atom in the Gen clusters strengthens ␲-type bond
delocalization.
As discussed before, when the seventh Ge atom in the
Ge7 ͑7a͒ is substituted by Cu, the stable CuGe6͑dЈ͒ isomer is
formed. Although the geometry is not distinctly distorted, the

bonding property is obviously changed. As seen from
HOMO of the Ge7 ͑7a͒, except for the second Ge and fourth
Ge atom, all the other Ge atoms participate in formation of
the ␴-type bond. However, for the substituted CuGe6͑dЈ͒, the
second and fourth Ge atoms begin to form the delocalized
␲-type bond with other Ge atoms.
As far as the large-sized Cu-concaved Gen clusters are
concerned, the ␴-type bond types among Ge atoms, which
are distributed beside Cu, usually change to be the delocalized ␲-type bonds. For example, as seen from HOMO of the
basketlike CuGe12 12cЈ, the obvious delocalized ␲-type
bond is formed between Cu and adjacent Ge atoms as compared to the identical Ge12 12a isomer.
FIG. 3. Contour maps of the HOMOs of the selected CuGen and Gen
clusters.

C. Effects of the doped Cu to bonding properties
of Gen clusters

As seen from the properties of the HOMO of the Ge2
dimer ͑Fig. 3͒, the ␲-type bond is formed between the Ge
and Ge atoms. However, when Cu is capped on the Ge2
molecule, the ␲-type bond is changed to be the ␴-type bond.
As observed from the HOMO of the Ge3 isomer, the delocalized ␲-type orbitals extend over all the three Ge atoms. As
mentioned above, two kinds of the CuGe3 geometries, which
have identical C2v symmetry, can be generated. However,
their shapes of the HOMO and bonding properties are obviously different. For the C2v͑aЈ͒ CuGe3, its HOMO obviously
corresponds to a ␴-type bond with mixed Cu d characters
and the density around the second Ge atom is very low.
However, for the C2v͑bЈ͒ CuGe3, its HOMO corresponds to
the delocalized ␲-type bond on the Ge3 unit with small admixture of the Cu d characters. According to the bonding
properties of the two CuGe3 isomers, it indicates that the

different evolution modes exist. As far as the C2v͑aЈ͒ CuGe3
isomer is concerned, it is generated from the substitution of
Cu on the Ge4 isomer. But, the C2v͑bЈ͒ CuGe3 isomer is
formed by capping Cu on the Ge3 C2v isomer. As observed
from HOMO of the Ge4 ͑4a͒, the 2Ge–4Ge bond appears the
␲-type bond and 1Ge–3Ge bond appears the ␴-type bond.
When the fourth Ge atom is substituted by Cu, the ␲-type
bond of 2Ge–4Ge disappears and the d-p ␴-type bond of
1Ge–Cu–3Ge is formed.
For the boatlike Ge6 6b isomer, the bonds of 1Ge–6Ge
and 3Ge–4Ge are obvious ␴-type bond with p subshell in
character; however, the density populated on the second Ge
and fifth Ge atoms is very low. When Cu is capped on the top
of the Ge6 cluster, the bonding properties are distinctly

D. Averaged binding energy, fragmentation energy,
and embedding energy of the Gen and CuGen
clusters

It is known that the relative stability of the different
sized clusters can be predicted by calculating the averaged
binding energy and fragmentation energy. Moreover, because
the doped Cu on Gen clusters can also influence the relative
stabilities of certain sized Gen clusters, the calculated results
on the averaged binding energy and fragmentation energy of
the Gen and CuGen clusters provide an interpretation of the
above influence.
The averaged binding energies and fragmentation energies for the Gen clusters can be defined as the following
formula:
Eb͑n͒ = ͓nET͑Ge͒ − ET͑Gen͔͒/n,

D͑n,n − 1͒ = ET͑Gen−1͒ + ET͑Ge͒ − ET͑Gen͒,
where ET͑Gen−1͒, ET͑Ge͒, and ET͑Gen͒ represent the total
energies of the most stable Gen−1, Ge, and Gen clusters, respectively. The averaged binding energies and fragmentation
energies for the Cu-doped CuGen clusters can be defined as
the following formula:
EbЈ͑n͒ = ͓ET͑Cu͒ + nET͑Ge͒ − ET͑CuGen͔͒/n + 1,
DЈ͑n,n − 1͒ = ET͑CuGen−1͒ + ET͑Ge͒ − ET͑CuGen͒,
where ET͑CuGen−1͒, ET͑Ge͒, ET͑Cu͒, and ET͑CuGen͒ represent the total energies of the most stable CuGen−1, Ge, Cu,
and CuGen clusters, respectively.
The calculated results on the averaged binding energies
are plotted as the curves which show the sized dependence of
the averaged binding energies for the CuGen clusters and Gen
clusters. As shown in Table II and Fig. 4, the averaged binding energy increases dramatically as the size of Gen changes
from 2 to 4; furthermore, the averaged binding energy in-

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Copper-doped germanium clusters

TABLE II. The averaged binding energies and fragmentation energies of the lowest-energy structure.
Cluster

BE ͑eV͒

FE ͑eV͒

Cluster

BE ͑eV͒

BEa ͑eV͒

BEb ͑eV͒

FE ͑eV͒

CuGe2
CuGe3
CuGe4
CuGe5
CuGe6
CuGe7
CuGe8
CuGe9
CuGe10
CuGe11
CuGe12
CuGe13

1.76
2.01
2.19
2.39
2.46
2.51
2.54

2.63
2.75
2.72
2.72
2.72

2.76
2.91
3.43
2.84
2.86
2.78
3.4
3.99
2.35
2.73
2.73

Ge2
Ge3
Ge4
Ge5
Ge6
Ge7
Ge8
Ge9
Ge10
Ge11
Ge12
Ge13

1.44
1.98
2.37
2.46
2.55
2.64
2.58
2.66
2.73
2.69
2.68
2.63

1.23
2.24
2.7
2.91
3.05
3.22
3.16
3.24
3.33
3.27
3.26
3.29

1.35
2.04
2.53

2.72
2.85
2.97
3.06
3.04
3.13
3.13
3.21
3.12

3.06
3.52
2.83
2.99
3.17
2.19
3.31
3.37
2.23
2.63
2.1

a

Theoretical binding energy per atom using DFT-GGA from Ref. 47.
Experimental binding energy per atom ͓for Ge2–8, measured atomization energy ͑Refs. 23, 48, and 49͒ for
Ge9–13, estimation from ion mobility ͑Ref. 50͔͒.
b

creases smoothly when the size of Gen clusters increases

from 4 to 10. However, when the size of Gen exceeds 10, the
averaged binding energy is gradually decreased as evinced
by drop of the stability. It reflects that the stabilities of the
small-sized Gen͑n ഛ 10͒ clusters have been enhanced with
increase of the size of clusters, and that the decrease of averaged binding energy of the large-sized Gen͑n Ͼ 10͒ clusters
predicts restriction of the formation of large-sized Gen clusters. The prediction about averaged binding energy of the
pure Gen clusters in this work is in agreement with the previous calculation results and experimental results.47,23,48–50
When Cu is doped on the pure germanium clusters, the
averaged binding energy is obviously influenced by Cu. As
shown in Fig. 4, the averaged binding energy of the
CuGen͑n ഛ 10͒ is lower than that of the Gen+1. In contrast,
the averaged binding energy of the encapsulated CuGen͑n
Ͼ 10͒ clusters is higher than that of identical-sized Gen clusters. For example, the averaged binding energy ͑2.718 eV͒ of
the encapsulated CuGe13 13bЈ is higher than that ͑2.631 eV͒
of the Ge13. It reveals that the encapsulation of Cu in the

large-sized Gen͑n Ͼ 10͒ increases its averaged binding energy and contributes to enhancing the stability of the caged
Gen clusters, and that the doped Cu atom elevates the averaged binding energy of the large-sized Gen clusters as compared to the pure Gen clusters. As can be seen from Fig. 4,
the averaged binding energy of the CuGen clusters increases
monotonously to maximum as the size of the CuGen clusters
increases from 1 to 10. However, the averaged binding energy of the CuGen clusters is gradually decreased as the size
of the CuGen exceeds 10.
On the other hand, the sized dependence of the fragmentation energies of the Gen and CuGen clusters is also investigated. As shown in Fig. 5, the fragmentation energy decreases with increase of the size of the small Gen clusters.
However, it increases when the size of Gen clusters is bigger
than 5. Our surprising finding is that the fragmentation energy sharply reaches to maximum at the Ge10 and decreases
dramatically after the Ge10 cluster, exhibiting that the Ge10 is
the most stable isomer. This finding is in good agreement
with the experimental measurement on the maximum ioniza-

FIG. 4. Sized dependence of the atomic binding energies of CuGen and Gen

͑n = 2 – 13͒ clusters.

FIG. 5. Sized dependence of the fragmentation energies of CuGen and Gen
͑n = 2 – 13͒ clusters.

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J. Chem. Phys. 123, 244303 ͑2005͒

J. Wang and J.-G. Han

tion potential of the Gen clusters being n = 10.25 Subsequently, the local abundant peaks of the D͑n , n − 1͒ appear at
the sizes of 7, 9, 10, and 12. However, when Cu is doped in
the Gen clusters, this situation is obvious. As can be seen
from Fig. 5, the local maxima of DЈ͑n , n − 1͒ appear at the
sizes of 5 and 10. Especially, the fragmentation energy of the
CuGe12 cluster is not distinctly higher than that of the
CuGe13 cluster. It indicates that the relative stability of the
CuGe12, which is different from Ta– Si12,44 is stronger than
that of the adjacent CuGe11, but is identical to the CuGe13
cluster. It should be pointed out that the binding energy and
fragmentation energy of the CuGe10 are the highest among
all sized clusters, indicating that the relative stability of the
CuGe10 cluster is stronger than other sized clusters, which is
in good agreement with the experimental observation on the
− 51
. As observed from the optimized CuGe10 structure,
CoGe10

all the Cu–Ge bond lengths and electrostatic interactions
among Cu and all the Ge atoms are almost equal. As for
basketlike CuGe12 12dЈ and CuGe13 13bЈ structures, some of
Ge atoms cannot efficiently interact with the Cu atom, which
are reflected from some Ge atoms being far from the concaved Cu atom; this finding implies that the number of dangling bonds of the Ge atoms in the basketlike structures is
more than that of the bicapped tetragonal antiprism or multirhombus CuGe10 10aЈ. Hence, it can be expected that the
encapsulated CuGe10 10aЈ cluster can be acted as building
block of the Cu-doped clustered-assembled materials because of the fourfold coordination of Ge atoms and no appreciable Ge dangling bonds. In addition, for all CuGen ͑n
Ͼ 9͒ clusters, the averaged binding energy and fragmentation
energy are significantly higher than those of the comparable
sized Gen clusters, and reflecting that the stability of CuGen
is enhanced when Cu is doped in the Gen clusters.
Beginning from the CuGe8, each sized Cu-doped cluster
has a stable encapsulated structure. In order to investigate the
stability of encapsulated structure with the variation of size,
it is necessary to calculate the embedding energy of the Cudoped Gen frameworks. And the embedding energy can be
defined as the following formula:

FIG. 6. Sized dependence of the embedding energies of CuGen ͑n = 8 – 13͒
clusters.

EE = ET͑Cu͒ + ET͑Gen͒ − ET͑CuGen͒,
where ET͑Cu͒, ET͑Gen͒, and ET͑CuGen͒ represent the total
energies of the most stable Cu, Gen and CuGen clusters, respectively.
As seen from Fig. 6, the embedding energy increases
rapidly when the size of the CuGen changes from 8 to 10,
then the embedding energy increases smoothly from 10 to
12. When size of CuGen varies from 12 to 13, the embedding
energy increases rapidly again. These findings indicate that
the extrastability of the encapsulated Cu atom in the small

Gen frames is not as strong as that of the encapsulated Cu
atom in the large Gen cages with increase of the size of the
CuGen clusters.
E. HOMO-LUMO gap, charge transfer, and
polarizability

Semiconducting characters of semiconductive material
germanium and the TM-doped germanium clusters can be
reflected from the energy gap between HOMO and LUMO.

TABLE III. The natural charge population, HOMO-LUMO gap, and dipole moment of the lowest-energy
structures of different sized CuGen and Gen ͑n = 2 – 13͒ clusters.

Cluster

Natural
population

HOMO-LUMO
gap ͑eV͒

Dipole
moment

Cluster

HOMO-LUMO
gap ͑eV͒

Dipole

moment

CuGe2
CuGe3
CuGe4
CuGe5
CuGe6
CuGe7
CuGe8
CuGe9
CuGe10
CuGe11
CuGe12
CuGe13

0.281
0.386
0.474
0.492
0.461
0.587
0.547
0.354
0.259
0.332
0.446
0.476

1.376
1.741

1.673
1.755
1.521
1.412
1.266
1.219
1.191
1.193
1.235
1.144

1.836
1.214
1.403
1.722
1.805
1.204
1.51
0.296
0.021
0.984
0.216
1.079

Ge2
Ge3
Ge4
Ge5
Ge6
Ge7

Ge8
Ge9
Ge10
Ge11
Ge12
Ge13

0.804
2.642
2.282
2.963
2.782
2.168
2.209
2.647
2.622
2.111
1.705
1.637

0
0.613
0
0.001
0.155
0.001
0.543
0.76
0.561
0.893

1.002
1.835

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244303-11

As seen from Table III, the HOMO-LUMO gaps of CuGen
are distinctly lower than those of Gen clusters.
Previous investigation suggested that charge-transferring
direction could be changed when the size of the TM-doped
Sin clusters ͑TM= Ni, Ta, and Zr͒ increases.32,44,52 However,
according to natural population analysis of the CuGen clusters, charge always transfers from Cu to the Ge atoms with
the increase of size, and indicates that Cu acts as electron
donor in all CuGen clusters. Local maximum of the charge
transfer and relative stability for the different sized CuGen
clusters can be found at 10; this finding provides a support of
our calculated relative stability by aid of the calculated fragmentation energy. It can be reflected that the charge transfer
in the encapsulated symmetrical CuGen is large and contributes to enhancing proportionate electrostatic interaction
which plays an important role in stabilizing the encapsulated
structures.
Wang et al.21 suggested that the dipole moments of Gen
͑n = 2 , 4 , 5 , 7͒ are nearly zero corresponding to high symmetry. In our calculation, similar results on the dipole moments
of the Gen clusters can be also obtained. Additionally, Wang
et al. also pointed out that polarizability of the Gen clusters
increases with decreasing HOMO-LUMO gaps; however, no
direct relationship between the polarizability and HOMOLUMO gap can be found for the CuGen clusters. For example, as seen from the optimized lowest-energy CuGe10 and
CuGe11 structures, the dipole moment of the CuGe10 is zero
because of the symmetric distribution of the Ge atoms
around Cu in the CuGe10 10a’ while the dipole moment of

the CuGe11 is close to 1 due to one Ge atom sticking out
from the CuGe11 11a’ structure; however, their HOMOLUMO gaps are almost identical. Hence, it can be expected
that the polarizabilities of the CuGen clusters are mainly dependent on symmetric distribution of the Ge atoms around
Cu. The dipole moment of the CuGen isomers generally decreases along with the increase of the dipole moment of the
Gen clusters as the size of the Gen and CuGen clusters increases, indicating that the Cu atom is doped in the center
sites of the Gen frames as the size of the CuGen clusters
increases.

IV. CONCLUSION

The growth behaviors, stabilities, electronic properties,
and polarizabilities of the Gen and CuGen ͑n = 2 – 13͒ clusters
are investigated theoretically at the UB3LYP level employing LanL2DZ basis sets. All the calculated results are summarized as follows.
͑1͒

J. Chem. Phys. 123, 244303 ͑2005͒

Copper-doped germanium clusters

According to optimized geometries of the Gen and
CuGen ͑n = 2 – 13͒ clusters, it is found that the growth
behaviors of the Cu-doped Gen clusters are different
from the pure Gen clusters. For the pure Gen clusters,
three different growth patterns, i.e., in-plane, out-ofplane edge-capped, or face-capped growth patterns, are
dominant for the different sized Gen clusters. However,
four different growth patterns, i.e., Cu-capped, Cusubstituted, Cu-concaved, and Ge-capped patterns, are
dominant for the different sized CuGen clusters.

͑2͒

͑3͒

͑4͒

According to the averaged binding energy analyses of
the Gen and CuGen clusters, it is concluded that the
doped Cu in the small-sized Gen ͑n Ͻ 10͒ clusters decreases the binding energies while the doped Cu in
large-sized Gen ͑n Ͼ 10͒ clusters increases the binding
energies. The calculated results on fragmentation energies of the Gen and CuGen clusters indicate that the
relative stabilities of the Cu-doped Gen ͑n ജ 10͒ clusters are enhanced as compared to the pure Gen clusters
͑n ജ 10͒. The magic numbers of the stabilities are 7, 9,
10, and 12 for the Gen clusters, and 5 and 10 for the
CuGen clusters. Furthermore, the relative stabilities of
the Cu-doped Gen ͑n = 5 , 10, 13͒ are obviously enhanced as compared to the identical-sized Gen clusters.
Moreover, the relative stability of the CuGe10 isomer
turns out to be the most stable cluster which is con−
firmed by the experimental measurement of the CoGe10
48
isomer.
Although the doped Cu in the Gen clusters does not
seriously distort the pure Gen frames, however, the
chemical bonding type of the HOMO is distinctly
changed as compared to the pure Gen clusters. It should
be mentioned that the analysis of the HOMO properties
contributes to explaining the growth behavior of the
CuGen clusters.
The HOMO-LUMO gap of the CuGen clusters decreases obviously as compared to the pure Gen clusters.
Unlike TM-Sin ͑TM= Ni, Zr, Ta, etc.͒, the charge in the
CuGen clusters always transfers from the Cu atom to
the Ge atoms. In addition, the relationship between the

polarizabilities and HOMO-LUMO gaps for the pure
Gen clusters is destroyed in the Cu-doped Gen clusters.

ACKNOWLEDGMENTS

This work is supported by National Natural Science
Foundation of China ͑20173055͒ and USTC fund.
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