Theoretical Study on the Structure and Energetics of Cd Insertion and Cu Depletion of CuIn 5 Se 8

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: [email protected] (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





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,


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,


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.


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





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.


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.


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.


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


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


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.


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


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


Fig. 9. All possible isomers and the energy differences, in kcal/mol, from the most stable isomers for MnXn211 clusters, n � 6±9. The



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.


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


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.


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.

References

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Johnson, M.A. Robb, J.R. Cheeseman, T.A. Keith, G.A.


Fig. 13. All possible isomers and the energy differences, in kcal/mol, from the most stable isomers for MnXn211 clusters, n � 13. The



Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-

Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J.

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Theoretical Study on the Structure and Energetics of Cd Insertion and Cu Depletion of CuIn 5 Se 8

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Transcript of Theoretical Study on the Structure and Energetics of Cd Insertion and Cu Depletion of CuIn 5 Se 8