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Theoretical study on the structure and energetics ofalkali halide clusters
M. Lintuluoto
Department of Environmental Information, Kyoto Prefectural University, Shimogamo, Sakyo-ku, Kyoto 606-8522, Japan
Received 12 June 2000; accepted 2 September 2000
Abstract
The structures of alkali halide clusters NanFn, LinFn and NanCln, and their metal-excess clusters NanF1n21; LinF1
n21 and
NanCln211 were investigated by the ab initio molecular orbital method for cluster sizes from 1 to 14. The magic numbers for
the neutral clusters NanFn, LinFn and NanCln are 4, 6, and 8. The most stable structure for these cluster sizes is a perfect crystallite
for NanFn and NanCln, and a double ring for LinFn. The magic numbers for the metal-excess clusters are 5 and 8, which are near
ideal cuboids with (100) facets. q 2001 Elsevier Science B.V. All rights reserved.
Keywords: NanFn; LinFn; NanCln; Neutral cluster; Positively charged cluster
1. Introduction
A cluster is a small piece of material containing
from three to several hundred atoms. Small clusters
show properties very different from those of bulk
material because of their small particle size. As cluster
size increases, there will appear structures where all
atoms can ®ll an ideally truncated lattice, so that there
are only perfect nanocrystalline surfaces. The nano-
surfaces of small clusters may show chemical proper-
ties different from those of bulk material, because of
their large surface-to-volume ratio and the large step
and defect density. There is the relation between the
cluster size and chemical properties, for example, a
particular cluster size is very reactive but the next size
is inert. An enough large cluster has the bulk-like
structure and chemical behavior. In addition to these
interests, studies on chemistry of clusters provide a
new way to study surface process. Defect sites, such
as steps, adatoms, and vacancies, play important roles
in solid-surface reactions [1±3]. Studies by using the
size-selected inorganic clusters may generate further
insight into the fundamental steps of complicated
chemical process on the defect surface as well as
those on the perfect surface.
Homer et al. reported the adsorption of NH3, H2O,
etc., which have a lone pair of electrons, to alkali
halide clusters [4,5]. The reactivity depends on the
cluster size and the alkali metal, and positively
charged clusters MnXn211 are much more reactive
than neutral and negatively charged clusters. KF,
NaF, and NaCl clusters have the same reactivity
patterns. Cuboidal nanocrystals (cluster sizes
n � 14, 23, 32, 38 and 53) of positively charged
clusters are nonreactive toward NH3 adsorption,
whereas basket-like clusters (n � 13 and 22), which
have defect sites, completely react with NH3.
Molecular NH3 has been suggested to ®ll the basket
defect. NanFn211 clusters are the most reactive, and the
LinFn211 cluster shows a different reactivity pattern
Journal of Molecular Structure (Theochem) 540 (2001) 177±192
0166-1280/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.
PII: S0166-1280(00)00741-7
www.elsevier.nl/locate/theochem
E-mail address: masami@unitary.kpu.ac.jp (M. Lintuluoto).
compared to other positively charged alkali halide
clusters.
Alford et al. reported ammonia chemisorption on
silicon cluster ions [6], and the reactivity depended on
the cluster size.
There have been many theoretical studies on the
relationship between the structure and cluster size of
metal halide clusters [7±15]. For examples, Martin
has studied the structures and properties of alkali
halide clusters by simple model calculation based on
classical electrostatic interactions [7], and Ochsenfeld
et al. have studied KnCln, LinFn and NanCln clusters by
using HF and MP2 methods [10±12] and alkali excess
clusters by HF method [14]. Barnett et al. studied the
water adsorption and reactions on small NaCl clusters
by the theoretical method, and they reported the water
molecule adsorbed at the halide-vacant (F-center) site
of the nonstoichiometric clusters [16]. However, there
have been still not many theoretical studies of the
adsorption and reaction on the clusters. As the struc-
tures of the clusters are closely related to their reac-
tivity patterns, the structure and energetics for each
cluster size of the metal halide clusters NanFn, NanCln
and LinFn and the metal-excess clusters NanFn211 ,
NanCln211 and LinFn21
1 by the ab initio method in this
study was investigated.
2. Model and calculational method
The alkali halide neutral and positively charged
clusters LiF, NaF and NaCl was examined. The
following basis sets were used for the Li, Na, F and
Cl atoms,
Li �10s�=�3s� set
1 polarization p function �zp � 0:076�
Na �11s5p�=�5s2p� set
1 polarization p function �zp � 0:061�
F �10s7p�=�3s2p� set
1 polarization d function �zd � 1:496�
Cl �11s8p�=�4s3p� set
1 polarization d function �zp � 0:514�
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192178
Fig. 1. All possible isomers and the energy differences, in kcal/mol, from the most stable isomers for MnXn clusters, n � 1±3. The dissociation
energy De of MnXn into nM1 and nX2 ions are shown for the most stable isomers in parentheses. All isomers are optimized at the HF level. The
white and shaded balls denote halide and metal atoms, respectively.
All of these basis sets were taken from Huzinaga
and Dunning [17,18] with a few modi®cations.
Hartree±Fock (HF) and second-order Mùller±Plesset
perturbation theory (MP2) calculations were
performed using the Gaussian 94 program [19].
The geometries of the structures were optimized by
the energy gradient method using the HF method, and
some structures were optimized using the MP2
method.
3. Structures and energetics of neutral clusters,MnXn
3.1. Cluster size n � 1±3
The structures of small clusters, with cluster sizes
of n � 1±3 was investigated. All of the possible
isomers were optimized at the HF level. The
optimized geometries of all possible isomers and the
energy differences from the most stable isomer calcu-
lated at the HF level are shown in Fig. 1. In this ®gure,
zero and larger values indicate the most and less stable
structures, respectively. The most stable structures of
LinFn and NanCln clusters are same as those reported in
the other theoretical studies by simple model calcula-
tion based on classical electrostatic interactions [7],
HF and MP2 methods [10,11] and local density func-
tional method [15].
The most stable structures were optimized with the
MP2 method. The geometries and Mulliken's atomic
charges of the most stable isomers calculated with the
MP2 method are shown in Table 1. The values from
the other theoretical studies are also collected in Table
1. The M±F distances, where M is an alkali metal
atom, in LiF and NaF clusters are 0.1 and 0.07 AÊ
longer than the corresponding experimental values
[20]. On the other hand, the calculated Na±Cl
distance in the NaCl cluster is in excellent agreement
with the experimental value of 2.360 AÊ .
Ring structures are the most stable isomers for all of
the M2X2 clusters, where X is a halogen atom. The
difference between ,XMX and ,MXM is 10.9, 2.3,
and 23.48 for the Li2F2, Na2F2 and Na2Cl2 clusters,
respectively.
Ring structures, which have the highest symmetry,
are the most stable isomers for all of the M3X3
clusters. For the Na3Cl3 cluster, the ring structure is
more stable than the double chain structure by only
0.3 kcal/mol using the MP2 method. Ochsenfeld et al,
also reported quite small value (1.2 kcal/mol by MP2
method)[11]. As shown in Fig. 1, only the Na3Cl3
cluster shows the double chain structure, those for
Li3F3 and Na3F3 are unstable. The MX bond lengths
in the M3X3 clusters are almost the same as those in
the M2X2 clusters. The difference between ,XMX
and ,MXM is 7.2 and 2.28 for the Li3F3 and Na3F3
clusters, respectively, whereas that for the Na3Cl3
cluster is 288.The Mulliken's atomic charges of the LinFn and
NanFn clusters are about ^0.9, whereas that of the
NanCln cluster is ^0.66±0.68. For clusters in this
size region, the atomic charge is not size-dependent.
The dissociation energy De of MnXn into M1
and X2 ions calculated by HF method are 178.9,
154.0 and 135.1 kcal/mol for LiF, NaF and NaCl
monomers, respectively, as shown in Fig. 1. The
dissociation energy calculated by MP2 method are
178.6 and 154.4 kcal/mol for LiF and NaF mono-
mers, respectively. These values are in good
agreement with the corresponding experimental
results [20] of 184.7 and 154.5 kcal/mol for LiF
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192 179
Table 1
The geometries and Mulliken's atomic charges of the most stable
isomers (n � 1±3) computed with the MP2 method (bond distances
and angles are in AÊ and degrees, respectively)
MXa r , MXMa Mulliken's
atomic charge
LiF r� 1.664 ^ 0.89
(exp.) r� 1.564
(ref. 10) r� 1.569 (MP2)
r� 1.564 (CCSD)
NaF r� 1.982 ^ 0.91
(exp.) r� 1.917
NaCl r� 2.356 ^ 0.68
(exp.) r� 2.360
(ref. 11) r� 2.361 (MP2)
Li2F2 r� 1.818 ^ 0.91
(exp.) r� 1.680
(ref. 10) r� 1.715, (MP2)
Na2F2 r� 2.141, ,MXM� 88.7 ^ 0.92
Na2Cl2 r� 2.541, ,MXM� 78.2 ^ 0.66
(ref. 11) r� 2.526, ,MXM� 78.4 (MP2)
Li3F3 r� 1.781, ,MXM� 116.0 ^ 0.91
Na3F3 r� 2.138, ,MXM� 119.0 ^ 0.93
Na3Cl3 r� 2.500, ,MXM� 105.8 ^ 0.66
a M and X� alkali metal and halogen atoms, respectively.
and NaF monomers, respectively. However, the
result for NaCl in this study is 154.0 kcal/mol
and is overestimated compared to the experimental
result [20] of 134.0 kcal/mol.
For the Na3Cl3 cluster, while the double chain struc-
ture has a dipole moment (8.21 D at the HF level), the
most stable structure (6-membered ring) has no dipole
moment. Furthermore, the ring structures for the
M2X2, Li3F3 and Na3F3 clusters, which are the most
stable structures for these clusters, also have no dipole
moments. We think that isomers with greater symme-
try and no dipole moment are the most favorable for a
cluster size of n � 2 or 3.
3.2. Cluster size n � 4; 5
The optimized geometries of all possible isomers
and the energy differences from the most stable
isomer calculated at the HF level are shown in Fig. 2.
The most stable structure of cluster size n � 4 for
all of the clusters is the cubic structure. The energy
difference between the ring and cubic structures for
the Li4F4 cluster is only 3.9 and 3.2 kcal/mol at the HF
and MP2 levels. These values are smaller compared to
the results by Ochsenfeld et al. [10], 9.5 and 10.4 kcal/
mol by HF and MP2 methods, respectively. The MX
bond lengths in the cubic structures are 1.92, 2.26, and
2.64 AÊ for the Li4F4, Na4F4 and Na4Cl4 clusters at the
MP2 level, respectively. When we compare the MX,
M3X3 and M4X4 clusters, there appears to be a rela-
tionship between the coordination number and the
bond length. The coordination numbers for MX,
M3X3 and M4X4 clusters are 1, 2 and 3, respectively.
As the coordination number of the metal and halide
clusters increases by one, the bond length increases by
about 0.15 AÊ . The atomic charges of the cubic
structures are ^0.89, 0.91 and 0.64 for the Li4F4,
Na4F4 and Na4Cl4 clusters, respectively. The atomic
charges do not change with the coordination number.
The most stable structure for the Na5Cl5 and Na5F5
clusters is the cubic structure, while the Li5F5 cluster
has two stable structures: the ring and cubic struc-
tures. The cubic structure is 2.8 and 4.6 kcal/mol
more stable than the ring structure at the HF and
MP2 levels, respectively.
In the previous section, isomers with greater
symmetry were more stable. However, the most
stable structure for the Na5F5 and Na5Cl5 clusters
has a low symmetry (Cs), while the symmetry of
the second-most stable structure is C4v. For NanXn
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192180
Fig. 2. All possible isomers and the energy differences, in kcal/mol, from the most stable isomers for MnXn clusters, n � 4±5. The dissociation
energy De of MnXn into nM1 and nX2 ions are shown for the most stable isomers in parentheses. All isomers are optimized at the HF level. The
white and shaded balls denote halide and metal atoms, respectively.
clusters of n � 4 and 5, structures that resemble a
portion of a crystal lattice are energetically more
favorable.
The dipole moments of the most stable structures
are smaller than those of other isomers with a cluster
size of n � 4 or 5. The dipole moments of the most
stable structures for the Na5Cl5 and Na5F5 clusters
calculated at the HF level are 3.23 and 3.21 D,
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192 181
Fig. 3. Assumed path of formation for each cluster size n � 1 to 5. The white and shaded balls denote halide and metal atoms, respectively.
respectively, while those of the second-most stable
structures are 8.31 and 8.38 D.
3.3. Increase in cluster size from n � 1 to 5
The increase in cluster size from n � 1 to 5 was
investigated at the HF level. With an increase from q
n � 1 to 2, we supposed that the two MX molecules
would bind to form an M2X2 cluster. This reaction has
no energy barrier. When the MX molecule adds to the
M2X2 cluster, the ring-structure M3X3 cluster is
formed with a small energy barrier (0.2 kcal/mol
calculated at HF method). There is a small energy
barrier (0.2 kcal/mol calculated at HF method) to
transform the double chain structure to the ring
structure for Na3Cl3, and the energy difference
between the two structures is very small (2.3 kcal at
the HF level). Therefore, we think that these two
structures are in thermal equilibrium.
For the increase from M3X3 to M4X4 clusters,
we suppose that the reaction has two steps, as
shown in Fig. 3. The same transformation from
M3X3 to M4X4 clusters was investigated by Martin
[7]. The ring structure M4X4 is formed with no
energy barrier. When the MX molecule forms
two M±X bonds with the M3X3 cluster, one M±
X bond of the M3X3 cluster spontaneously cleaves
to form the ring-structure M4X4 cluster. In the
next step, four atoms are considered to lie in the
same plane, while the remaining four atoms lie in
a parallel plane. The center plane is constrained
and the two other planes bend in the same direc-
tion until the cubic structure is formed, as shown
in Fig. 3. The angle between the center and other
bending planes is ®xed, and all of the bond
lengths are optimized at each point in the energy
diagrams in Fig. 4. The energy barriers in the
second step are 17.8, 6.3 and 6.3 kcal/mol for
the LiF, NaF and NaCl clusters, respectively.
These barriers are quite small. In the LiF cluster,
the energy difference between the ring and cubic
structures is also small (3.9 kcal/mol). Therefore,
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192182
Fig. 4. Energy pro®les for the formation of the cubic structure from the ring structure. u indicates the angle between the center and the other
bending plane. The white square, black triangle and white circle denote the Li4F4, Na4F4 and Na4Cl4 clusters, respectively.
we think that these two structures are in thermal
equilibrium.
The MX bond of the M4X4 cluster spontaneously
cleaves to form the M5X5 cluster with no energy
barrier, as is the case with clusters of other sizes.
3.4. Cluster size n � 6±9
The most stable structure for LinFn clusters of sizes
of n � 6, 8 and 9 is the double ring structure, while the
most stable structure for Na6F6 and Na6Cl6 clusters is a
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192 183
Fig. 5. All possible isomers and the energy differences, in kcal/mol, from the most stable isomers for MnXn clusters, n � 6±9. The dissociation
energy De of MnXn into nM1 and nX2 ions are shown for the most stable isomers in parentheses. All isomers are optimized at the HF level. The
white and shaded balls denote halide and metal atoms, respectively.
cubic structure, like a portion of a perfect crystallite,
as shown in Fig. 5. The energy differences between
the double ring and cubic structures are 2.8, 2.2 and
0 kcal/mol for the Li6F6, Na6F6 and Na6Cl6 clusters at
the MP2 level, respectively. The double ring structure
is more stable than the cubic structure for the Li6F6
cluster at the MP2 level as same as that at the HF
level. Ochsenfeld et al reported 3.9 and 3.7 kcal/mol
for the Li6F6 and Na6Cl6 clusters at the MP2 level,
respectively, and the double ring structure is still
more stable than the cubic structure for the Li6F6 clus-
ter [10,11]. The energy difference between the double
ring and cubic structures for the Li8F8 cluster is also
quite small (1.4 kcal/mol at the MP2 level).
Structures that resembled a portion of a crystal
lattice were energetically more favorable at cluster
sizes of n � 4 and 5 in the case of NanFn and NanCln
clusters. The same tendency was also noted for NanFn
and NanCln clusters at cluster sizes from n � 6 to 9.
The energy differences between the double- or triple-
ring and lattice structures for the LinFn cluster
decreased from cluster size n � 6.
Fig. 6 shows the energy required to remove LiF,
NaF or NaCl from the most stable clusters of
different sizes. The LiF, NaF or NaCl is removed
from each of these clusters, and then the relaxation
of remaining structures follows. These energy
include the relaxation energy of remaining struc-
tures. It is easy to identify stable cluster sizes
using Fig. 6. For the NaF and NaCl clusters, clus-
ters sizes of n � 4, 6 and 8 are local maxima (i.e.
more stable) and n � 3, 5 and 7 are local minima.
For the NaF and NaCl clusters, cluster sizes n � 4,
6 and 8 show the perfect crystallites. The results
for the LinFn cluster are somewhat different. For
the LinFn cluster, while cluster size n � 6, which
has a double-ring structure, has a maximum value,
cluster sizes n � 4 and 8 are not.
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192184
Fig. 6. The energy required to remove LiF, NaF or NaCl from the most stable MnXn (n � 2±9; 12±14) clusters. The white square, black triangle
and white circle denote the LinFn, NanFn and NanCln clusters, respectively.
3.5. Cluster size n � 12±14
The isomers for cluster sizes n � 12±14 are shown
in Fig. 7. The structures are optimized only at the HF
level; we did not calculate at the MP2 level, due to
computational dif®culties.
The triple-ring structure is the most stable structure
for the Li12F12 cluster, while the perfect crystallite is
the most stable structure for the Na12F12 and Na12Cl12
clusters. The perfect crystallite is the most stable
structure for LinFn clusters beginning at a cluster
size of n � 13.
All of the clusters show local minima at a cluster
size of n � 13 in Fig. 6. The M13X13 cluster has a point
defect structure; i.e. a metal de®cit. M12X12 has a
perfect lattice structure, while M14X14 has a stepped
lattice structure. Thus, a point defect is less stable than
a stepped or perfect lattice.
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192 185
Fig. 7. All possible isomers and the energy differences, in kcal/mol, from the most stable isomers for MnXn clusters, n � 12±14. The
dissociation energy De of MnXn into nM1 and nX2 ions are shown for the most stable isomers in parentheses. All isomers are optimized at
the HF level. The white and shaded balls denote halide and metal atoms, respectively.
4. Structures and energetics of positively chargedclusters, MnXn21
1
4.1. Cluster size n � 2±5
We next examine the structures and energies of
small positively charged clusters MnXn211 for n � 2
to 5. Compared to neutral clusters MnXn, these clusters
are missing one halide anion, and thus have a positive
charge.
The structures of all isomers optimized at the HF
level are shown in Fig. 8. A three-dimensional struc-
ture is the most stable for the Na3Cl21 cluster, while a
chain structure is the most stable for Li3F21 and Na3F2
1.
Three-dimensional structures are the most stable for
M4X31 clusters, while the most stable structures for
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192186
Fig. 8. All possible isomers and the energy differences, in kcal/mol, from the most stable isomers for MnXn211 clusters, n � 2±5. The
dissociation energy De of MnXn211 into nM1 and (n 2 1)X2 ions are shown for the most stable isomers in parentheses. All isomers are
optimized at the HF level. The white and shaded circles denote halide and metal atoms, respectively.
M5X41 clusters are almost planar. The energy differ-
ences between the most stable structures and other
isomers are large, except for Na3Cl21. In these cases,
the most stable structures are likely dominant.
However, the three isomers for the Na3Cl21 cluster
exist in almost equal proportions. The electron corre-
lation by the MP2 method does not change the order
of the stable isomers or the energy differences.
The geometries and Mulliken's atomic charges of the
most stable isomers calculated with the MP2 method are
shown in Tables 2 and 3, respectively. The atomic
charges of the LinFn211 and NanFn21
1 clusters are almost
the same, while those of the NanCln211 clusters are
smaller than those of other clusters. There is a relation
between the coordination number and the atomic charge
on the metal atoms for all of the clusters MnXn211 : qM1 .
qM2 . qM3.
4.2. Cluster size n � 6±9
The isomers calculated at the HF level for cluster
sizes of n � 6 to 9 are shown in Fig. 9. For M6X51
clusters, one halide atom has been removed from the
most stable M6X6 neutral clusters, which have a
double 6-membered ring structure. For the Li7F61 clus-
ter, one ¯uoride atom has been removed from the most
stable Li7F7 cluster, so one lithium atom is put on the
double 6-membered ring structure. For the Na7F61 and
Na7Cl61 clusters, the metal atoms are located at the
center of 6-membered rings, since the diameters of
both 6-membered rings are large enough to accommo-
date the Na atom. However, the 6-membered ring of
Li7F61 is too small for the Li atom. The M8X7
1 clusters
have a lattice-like structure, which is like a stepped
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192 187
Table 2
The geometries of the most stable isomers MnXn211 (n � 1±3)
computed with the MP2 method (bond distances rMc±Xc (The bond
distance between the metal (Mc) and halide atoms (Xc)) and angles
aXcMcXc (The bond angle Xc±Mc±Xc), aMcXcMc (The bond angle
Mc±Xc±Mc) are in AÊ and degrees, respectively; c indicates the
coordination number of the metal or halide atom)
MX Bond length/angle
Li2F1 rM1±X2� 1.752
Na2F1 rM1±X2� 2.076
Na2Cl1 rM1±X2� 2.464
Li3F21 rM2±X2� 1.787, rM1±X2� 1.729
Na3F21 rM2±X2� 2.171, rM1±X2� 2.050
Na3Cl21 rM2±X3� 2.652/aX3M2X3� 81.5,
aM2X3M2� 82.1
Li4F31 rM2±X3� 1.872, rM3±
X3� 1.938/aX3M3X3� 87.3,
aX3M2X3� 91.2,
aM3X3M2� 86.5
Na4F31 rM2±X3� 2.201, rM3±
X3� 2.283/aX3M3X3� 83.9,
aX3M2X3� 87.9,
aM3X3M2� 89.1
Na4Cl31 rM2±X3� 2.593, rM3±
X3� 2.652/aX3M3X3� 93.0,
aX3M2X3� 95.8,
aM3X3M2� 83.7
Li5F41 rM2±X3� 1.844, rM4±
X3� 1.949/aX3M4X3� 90.0,
aX3M2X3� 96.6,
aM4X3M2� 86.7
Na5F41 rM2±X3� 2.201, rM4±
X3� 2.334/aX3M4X3� 90.0,
aX3M2X3� 97.2,
aM4X3M2� 86.4
Na5Cl41 rM2±X3� 2.555, rM4±
X3� 2.791/aX3M3X3� 90.0,
aX3M2X3� 97.4,
aM3X3M2� 86.3
Table 3
The Mulliken's atomic charges on the atoms (Mc and Xc) (the
Mulliken's atomic charge on the atom with coordination number
c) of the most stable isomers MnXn211 (n � 1±3) computed with the
MP2 method
MX Mulliken's atomic charge
Li2F1 M1�10.97, X2�20.93
Na2F1 M1�10.97, X2�20.94
Na2Cl1 M1�10.84, X2�20.68
Li3F21 M2�10.94,
M1�10.96, X2�20.93
Na3F21 M2�10.94,
M1�10.97, X2�20.94
Na3Cl21 M2�10.72,
M1�10.81, X2�20.67
Li4F31 M3�10.89,
M2�10.96, X3�20.92
Na4F31 M3�10.88,
M2�10.96, X3�20.92
Na4Cl31 M3�10.63,
M2�10.76, X3�20.63
Li5F41 M4�10.92,
M2�10.94, X3�20.93
Na5F41 M4�10.92,
M2�10.96, X3�20.94
Na5Cl41 M4�10.56,
M2�10.76, X3�20.65
lattice surface. The M9X81 clusters have a structure in
which one metal atom is added to 8-membered ring
structures, while the Na9X81 cluster has a structure in
which one halide atom is removed from the most
stable Na9X9 cluster.
The energy required to remove a metal cation and
LiF, NaF or NaCl from the most stable clusters of
different sizes is shown in Figs. 10 and 11, respec-
tively. The metal cation and LiF, NaF or NaCl is
removed from each of these clusters, and then the
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192188
Fig. 9. All possible isomers and the energy differences, in kcal/mol, from the most stable isomers for MnXn211 clusters, n � 6±9. The
dissociation energy De of MnXn211 into nM1 and (n 2 1)X2 ions are shown for the most stable isomers in parentheses. All isomers are
optimized at the HF level. The white and shaded circles denote halide and metal atoms, respectively.
relaxation of remaining structures follows. It is easy to
determine the stable cluster size using Figs. 10 and 11.
Maximum values indicate that clusters of those sizes
are more stable than those of other sizes.
Each cluster shows different features in Fig. 10. The
maximum values for the LinFn211 , NanFn21
1 and
NanCln211 clusters, are at n� 8, 5 and 6, respectively.
Na7F61 and Na7Cl6
1 are minima and Li7F61 is a local
minimum. The structure that remains after the metal
cation is removed from M7X61 is the double 6-
membered ring structure. Since the double 6-
membered ring structure is stable for all types of
clusters, little energy is needed to remove the metal
cation.
On the other hand, Fig. 11 is almost the same as Fig.
10. The minimum and local minimum points are at
cluster sizes of n � 3 and 7 for all of the clusters and
the maximum and local maximum points are at cluster
sizes of n � 8 and 5, respectively, for all of the clusters.
4.3. Cluster size n � 12±14
The isomers of cluster sizes of n � 12 to 14 opti-
mized at the HF level are shown in Figs. 12 and 13.
The most stable structures for the M12X111 and
M14X131 clusters are shown in Fig. 12. For all of the
clusters, the most stable structure is a lattice at a
cluster size of n � 14.
Basket structures, which have a hole inside the
clusters, are the most stable for cluster size n � 13,
as shown in Fig. 13. However, the energy difference
from the second-most stable structure is quite small
for the Na13Cl121 cluster, 1.1 kcal. Homer [4,5] et al.
support the basket structure. They think that this
structures is highly reactive for the adsorption of
molecular NH3, which may be adsorbed into the
hole of the basket structure.
Fig. 11 shows the energy required to remove a
metal cation from the most stable clusters of different
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192 189
Fig. 10. The energy required to remove a metal cation from the most stable MnXn211 clusters (n � 2±9; 12±14). The white square, black triangle
and white circle denote the LinFn211 , NanFn21
1 and NanCln211 clusters, respectively.
M. Lintuluoto / Journal of Molecular Structure (Theochem) 540 (2001) 177±192190
Fig. 11. The energy required to remove LiF, NaF or NaCl from the most stable MnXn211 clusters (n � 2±9). The white square, black triangle and
white circle denote the LinFn211 , NanFn21
1 and NanCln211 clusters, respectively.
Fig. 12. The most stable isomers and their dissociation energy De in kcal/mol for MnXn211 clusters, n � 12 and 14. The isomers are optimized at
the HF level. The white and shaded circles denote halide and metal atoms, respectively.
sizes. M14X131 clusters, which have a lattice structure,
are quite stable compared to clusters of other sizes.
Cluster size n � 14 has been shown to be a magic
number for LinFn211 experimentally [21].
5. Summary
Their magic numbers of all smaller neutral clusters,
n , 9, are 4, 6 and 8. A smaller LinFn neutral cluster,
n , 9; shows different characteristics from similar
sized NanFn and NanCln clusters. The most stable
structures at magic numbers, n � 4, 6 and 8 are the
perfect crystallite for the NanFn and NanCln clusters
and the double ring structure for the LinFn cluster.
The perfect crystallite is also the most stable
structure for the LinFn cluster starting at cluster size
n � 12.
The energy barrier between the ring and cubic
structures for cluster size n � 4 is 6.3 kcal/mol for
Na4F4 and Na4Cl4 clusters, while it was 17.8 kcal/
mol for the Li4F4 cluster.
The most stable structures for each cluster size are
almost the same for the three metal-excess clusters
NanFn211 , LinFn21
1 and NanCln211 . The magic numbers
are 5 and 8, and their structures are close to ideal
cuboids with (100) facets. Basket structures are the
most stable for cluster size n � 13 for the three
metal-excess clusters Na13F121, Li13F12
1 and Na13Cl121.
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