ORI GIN AL PA PER

Density Functional Study of Structural and ElectronicProperties of AlPn (2 £ n £ 12) Clusters

Ling Guo

Received: 4 October 2012 / Published online: 6 December 2012

� Springer Science+Business Media New York 2012

Abstracts Low-lying equilibrium geometric structures of AlPn (n = 2–12) clus-

ters obtained by an all-electron linear combination of atomic orbital approach,

within spin-polarized density functional theory, are reported. The binding energy,

dissociation energy, and stability of these clusters are studied within the local spin

density approximation (LSDA) and the three-parameter hybrid generalized gradient

approximation (GGA) due to Becke–Lee–Yang–Parr (B3LYP). Ionization poten-

tials, electron affinities, hardness, and static dipole polarizabilities are calculated for

the ground-state structures within the GGA. It is observed that symmetric structures

with the aluminum atom occupying the peripheral position are lowest-energy

geometries. And the Al impurity in the most stable structures of AlPn clusters can be

looked upon as a substitutional impurity in pure Pn?1 clusters or capping Al atom in

the different peripheral positions of pure Pn clusters. Generalized gradient

approximation extends bond lengths as compared to the LSDA lengths. The odd–

even oscillations in the dissociation energy, the second differences in energy, the

HOMO–LUMO gaps, the ionization potential, the electron affinity, and the hardness

are more pronounced within the GGA. The stability analysis based on the energies

clearly shows the AlP5 and AlP7 clusters to be endowed with special stabilities.

Keywords Aluminum phosphide � DFT theory � Stability

Introduction

Small clusters composed of phosphorus atom have been the subjects of intensive

studies for the last two decades. A large number of studies of phosphorus clusters,

both theoretical as well as experimental have been reported (See, for example, the

L. Guo (&)

School of Chemistry and Material Science, Shanxi Normal University, Linfen 041004, China

e-mail: gl-guoling@163.com

123

J Clust Sci (2013) 24:165–176

DOI 10.1007/s10876-012-0539-y

reviews in Refs. [1–3].) One of the main motivations behind these studies is to

understand the evolution of physical properties with the size of the cluster. Many

properties of phosphorus clusters can be understood using the spherical jellium

model (SJM) [4], in which the ions are smeared out in a uniformly charged sphere

leading to electronic shell closures for clusters containing a ‘magic’ number 2, 8, 20,

40, 58, 92, 138, … of valence electrons. These findings were subsequently

confirmed by first-principles theoretical calculations in which the ions were

represented [5]. The question we address here is the effect of doping by a single

impurity on the electronic structure and geometry of these clusters. In bulk

materials, a small percentage of impurity is known to affect the properties

significantly. In clusters, the impurity effect should be even more pronounced and

influenced by the finite size of the system. Taylor and coworkers [6] performed

some experiments in this direction. They reported the experimental adiabatic

electron affinity and vertical detachment energy of AlPn (n = 1–4).

This experimental work triggered an interest in simulations of Al doped

phosphorus clusters. Archibong et al. [7, 8] have reported the equilibrium

geometries, harmonic vibrational frequencies and electron detachment energies of

the neutral and anion AlP2 and AlP3 performed at density functional theory

(B3LYP, BP86 and BPW91-DFT) and ab initio methods [MP2 and Coupled Cluster

Singles and Doubles with Connected Triples CCSD(T)]. Feng and Balasubramanian

[9, 10] have also studied the structures and potential energy curves of electronic

states of AlP2, AlP3 and its ions, using the complete active space self-consistent

field (CASSCF) method followed by multireference singles and doubles configu-

ration interaction (MRSDCI). Other theoretical and experimental studies on AlPn

have been also published [11, 12].

To provide further insight on AlPn clusters, we have carried out a detailed

systematic study of the equilibrium structure and various electronic-structure related

properties of these clusters, employing both the local spin density (LSDA) and

hybrid generalized gradient approximation (GGA) for the exchange–correlation

potential. We investigate the relative ordering of these structures with the Al

impurity occupying the peripheral and other different position, and show that the

ground-state structures have Al taking a peripheral position. The calculations are

explicitly carried out, to our knowledge for the first time, by considering all

electrons in the calculations with no pseudopotentials (with nonlocal gradient

corrections). Furthermore, the all-electron treatment eliminates issues like core–

valence exchange–correlation, which occurs in the pseudopotential treatment when

there is no marked distinction between the core and valence regions [13]. Here, we

study the evolution of the ionization potential, electron affinity, HOMO–LUMO

gap, hardness, polarizability, dissociation energy, and binding energy for AlPn

clusters up to n = 12. These physical quantities are compared with their

counterparts calculated at the same level (all-electron B3LYP/6-311?G*) for pure

phosphorus clusters, which to our knowledge also represent the first all-electron

with gradient corrections calculations in these systems. By compared with previous

calculations, our B3LYP results about AlP2 and AlP3 are close to the earlier

MRSDCI?Q and CCSD(T) results[7–10]. In addition, with the methods and basis

sets, we obtain the geometries of AlPn (n = 4–12) for the first time.

166 L. Guo

123

In the following section, we briefly outline the computational methodology. In

‘‘Methodology and Computational Details’’ section the results are presented and

discussed, and we conclude in ‘‘Results and Discussion’’ section.

Methodology and Computational Details

The geometry optimization and electronic-structure calculation is carried out using

a molecular-orbital approach within the framework of spin-polarized density

functional theory [14, 15]. An all-electron 6-311?G* basis set was employed [16,

17]. We have employed KS exchange along with the Vosko et al. [18]

parameterization of homogeneous electron gas data due to Ceperley and Alder

[19] for the correlation potential. We shall henceforth refer to this specific LSDA

approach as SVWN. We have also carried out calculations that go beyond the

LSDA and take into account gradient corrections. In this case, we have used

Becke’s three parameter functional (B3LYP), [20] which uses part of the Hartree–

Fock exchange (but calculated with KS orbitals) and Becke’s [21] exchange

functional in conjunction with the Lee et al. [22] functional for correlation. The

ionic-configuration was regarded as optimized when the maximum force, the root

mean square (rms) force, the maximum displacement of atoms, and the rms

displacement of atoms have magnitudes less than 0.0045, 0.0003, 0.0018, and

0.0012 a.u., respectively. We carried the calculations out for spin multiplicities of

2S ? 1 = 1 and 2S ? 1 = 2 for clusters with even and odd numbers of electrons,

respectively. All calculations are carried out using GAUSSIAN 98 [20] suite of

programs.

Results and Discussion

The geometries of all the clusters obtained within the LSDA and B3LYP are similar,

apart from the larger bond distances observed in B3LYP, although the order of

isomers is reversed in some cases with B3LYP. We therefore present only the

structures obtained within the B3LYP scheme in Fig. 1.

In the B3LYP scheme, the lowest-energy structure for AlP2 is an isosceles

triangular structure, with the Al atom at the apex, which is similar to elemental

trimers P3 [23]. Feng and Balasubramanian [9] reported a theoretical bond lengths

of 2.599 and 1.989 A for Al–P and P–P bonds and a bond angle of 45.0� at the

MRSDCI?Q level of theory with relativistic effective core potentials (RECPS) and

3s3p valence basis sets. Achibong et al. [7] have optimized the geometry with

rAl–P = 2.603 A, rp–p = 1.985 A, hPAlP = 44.8� at the BPW91 level, and

rAl–P = 2.580 A, rp–p = 1.990 A, hPAlP = 45.4� at the CCSD(T) level with the

6-311?G(2df) one-particle basis set. Our B3LYP results are close to the earlier

MRSDCI?Q and CCSD(T) results. A bent chain with Cs (2A00) symmetry in which

Al takes a terminal position, and a linear structure in which Al takes central

position, are two low-lying structures at, respectively, 0.66 and 2.66 eV above the

most stable structure.

Density Functional Study 167

123

Two energetically degenerate structures are found for AlP3 in the B3LYP

scheme. AlP3 is a stable cluster, and many experimental and theoretical studies have

been reported. Liu et al. [12] have observed the AlP3- cluster in TOF. Gomez et al.

[6] reported the experimental adiabatic electron affinity (2.06 ± 0.05 eV) and

vertical detachment energy (2.58 ± 0.025 eV) for AlP3. The previous theoretical

studies of the AlP3 geometry include the 1999 work by Feng and Balasubramanian

[10] at the ab initio CASSCF/MRSDCI level of theory with the RECPs?3s3p basis

sets, and the 2002 work by Archibong et al. [8] with the B3LYP-DFT, MP2 and

CCSD(T) method. Feng’s studies appeared to have established the ground state

geometry of AlP3 to be the pyramidal C3v (3A2) structure. We give the different

Fig. 1 Geometries of AlPn structures

168 L. Guo

123

conclusion. The most stable one, like the valence isoelectronic AlAs3 [24], has a

rhomboidal (C2v, 1A1) structure which can be obtained by substituting a P atom by

an Al atom in the P4- anion [23]. This singlet state is lower in energy by at least

0.5 eV than the triplet (3A2–C3v) state previous predicted by Feng et al. [10]. And

our optimized AlP3 ground state is consistent with Archibong et al.’s results [8]. The

other energetically degenerate structure is a Cs (1A0) isomer and is above the most

stable one by 0.08 eV.

In the case of AlP4, three low-lying nearly degenerate structures are found, two of

which are 3-D structure and the other one is planar. The ground state of the AlP4

molecule is found to be the 3-D (C2v, 2A1) isomer (a in Fig. 1), which is similar to

P5 [23] and the valence-isoelectronic AlAs4 [25], and this similar proves the Gomez

et al.’s [6] prediction that small AlP clusters adopt the two- and three-dimensional

characteristic of GaxAsy clusters. There exists two kinds of P–P and one kind of

Al–P bonds in the neutral ground state, and the Al–P bond lengths are longer than

those for axial and equatorial P–P bonds by about 0.178 and 0.246 A, respectively.

A pentagon planar structure (b in Fig. 1) is higher by 0.25 eV, which can be

considered to replace a P atom in an apical position with a Al atom in the structure

of P5 (a planar D5h pentagon). And the square pyramid (C4v, 2A1) (c in Fig. 1) is

above the lowest energy structure by 0.65 eV.

The AlP5 ground state has the high C5v (1A1) symmetry, which is derived from

the P5 cluster by placing a fivefold Al atom on the top. The other low-lying structure

with the lower Cs (1A0) symmetry is a distorted triangle prism lying 1.05 eV higher

in energy. The Al impurity in this geometry can be looked upon as the substitutional

impurity in the low-lying triangle prism P6 cluster.

The equilibrium lowest-energy geometry of AlP6 is derived from a boat-shape P6

by adding of one twofold Al atom between two P atoms. A face-capped triangle

prism (b in Fig. 1) is higher than the lowest-energy structure by 0.38 eV, which can

be viewed as capping an additional Al atom on the square face of the triangle prism

P6.

The present calculations consider a cuneane structure as the ground state of AlP7

cluster, which can be derived from a square-face-capped triangle prism P7 by adding

an additional threefold Al atom. The symmetry of P7 is changed from C2v to lower

Cs (1A0) symmetry of AlP7 in the procession. Next AlP7 isomer in the energy

ordering is a distorted cube structure (b in Fig. 1) with Cs (1A0) symmetry lying

0.98 eV above the ground state, which is obtained by substitution of one P atom by

one Al atom in the cube P8.

The most stable geometry of AlP8 is the result of the addition of the Al atom to

the lowest-energy structure of P8 cluster. The Cs symmetry structure with Al

impurity at its center is higher by 0.21 eV. The lowest-energy state of AlPn

structures for n [ 8 are also results of substitutional impurity in pure Pn?1 clusters

or capping the Al impurity in the different positions of Pn clusters.

Thus our all-electron spin-polarized results show that the Al impurity prefers a

peripheral position. In general, the Al impurity in the most stable structures of AlPn

clusters can be looked upon as a substitutional impurity in pure Pn?1 clusters or

cappinging Al atom in the different peripheral positions of pure Pn clusters.

Density Functional Study 169

123

We now discuss the relative stability of these clusters by computing the energetic

that are indicative of the stability. We compute the atomization or binding energy

(BE) per atom, the dissociation energy (DE), and the second differences of energy

as, respectively,

Eb AlPn½ � ¼ nE P½ � þ E Al½ � � AlPn½ �= nþ 1ð Þ; ð1ÞDE AlPn½ � ¼ E AlPn½ � � E AlPn�1½ � � E P½ �; ð2Þ

D2E AlPn½ � ¼ AlPnþ1½ � þ E AlPn�1½ � � 2E AlPn½ � ð3ÞThe calculated binding energies are shown in Fig. 2. The binding energy

increases rapidly from 3.19 eV (3.3 eV with LSDA) for AlP2 to 4.22 eV (4.56 eV

with LSDA) for AlP7, with a small peak at AlP7, and beyond it tends to saturate. The

LSDA binding energies are larger than the B3LYP binding energies. This trend is

consistent with the generally observed overbinding tendency within LSDA. The BE

curve of pure phosphorus clusters calculated at the B3LYP/6-311?G* level of

theory is also shown in the same figure. Its comparison with the BE curve for AlPn

clusters show that the small clusters of AlPn are weakly bound. As the cluster grows

in size, the difference between the BE curves of AlPn clusters and pure phosphorus

clusters stead diminishes, and the bonding in doped clusters would be essentially

similar to that in pure clusters. It is also seen from Fig. 2 that odd–even effects are

more prominent in doped clusters than in pure clusters.

The calculated values of the energy required for dissociation of AlPn into AlPn-1

and P are presented in Fig. 3. The LSDA values of dissociation energy are higher

than their B3LYP counterpart, again indicating its overbinding nature. The curve

shows odd–even oscillations with a peak for clusters with an even number of

electrons except for AlP10 cluster. The peak at AlP10 within the LSDA is especially

prominent.

The values of second difference of cluster energy, Eq. (3), are plotted in Fig. 4.

This curve again shows odd–even oscillation within both B3LYP and LSDA

methods, with peaks (dips) at an odd (even) number of phosphorus atoms. This is so

because clusters with an odd n present closed-shell states. However, the peak at

AlP10 is conspicuous in LSDA. This observation is consistent with that from the

dissociation energy, indicating that the LSDA, the AlP10 cluster is more stable.

We have also calculated the adsorption energy of Al, i.e., the energy released

upon adsorption of Al by a pure phosphorus cluster, according to

3.13.33.53.73.94.14.34.54.74.9

2 4 6 8 10 12 14

No.of atoms

Bin

ding

ene

rgy

per

atom

(eV

)

Fig. 2 Binding energy inelectron volt. Solid line B3LYP,dashed line SVWN; and thedotted dashed line Pn with theB3LYP

170 L. Guo

123

Ead ¼ E AlPn½ � � E Pn½ � � E Al½ � ð4ÞThe calculated values of Ead for the clusters up to P12 ranges between 1.41 and

3.72 eV (Table 1). The minimum value (1.41 eV) occurs for AlP12, while it takes

the maximum value (3.72 eV) for AlP3.

The HOMO–LUMO gap is a useful quantity for examining the stability of

clusters. It is found that systems with larger HOMO–LUMO gaps are, in general,

less reactive. In the case of an odd-electron system, we calculate the HOMO–

LUMO gap as the smallest spin-up-spin-down gap. The HOMO–LUMO gaps as

thus calculated are presented in Fig. 5. The HOMO–LUMO curve shows the same

behavior in the LSDA and B3LYP schemes. We note that the HOMO–LUMO gaps

present a similar oscillating behavior as observed for D2E[AlPn]. Clusters with an

even number of electrons have a larger HOMO–LUMO energy gap and therefore

are expected to be less reactive than clusters with an odd number of electrons. As

already mentioned, the stability exhibited by even number of electrons clusters is

due to their closed-shell configurations that always comes along with an extra

stability. It is important to mention that this result is agreement with the electronic

shell jellium model [4], where filled-shells cluster with 2, 8, 18, 20, 40, 58, 92, …valence electrons have increased stability, the mass spectra of cluster distribution

33.5

44.5

55.5

66.5

7

2 4 6 8 10 12 14

No.of P atoms

Dis

soci

atio

n en

ergy

(eV

)

Fig. 3 Dissociation energy inelectron volt. Solid line B3LYP.Dotted line SVWN

-3

-2

-1

0

1

2

3

2 4 6 8 10 12

No.of P atoms

Sec

ond

diffe

renc

e in

ener

gy (

eV)

Fig. 4 Second difference inenergy. Solid line B3LYP.Dotted line SVWN

Table 1 Adsorption energies (in eV) (See text for full details) calculated within B3LYP with

(6-311?G*) basis set

n 2 3 4 5 6 7 8 9 10 11 12

Ead 1.46 3.72 1.43 3.39 2.96 3.30 1.81 1.72 1.96 2.38 1.41

Density Functional Study 171

123

shows pronounced intensity in clusters with these number of atoms, the so-called

magic numbers.

Experimentally, the electronic structure is probed via measurements of ionization

potentials, electron affinities, polarizabilities, etc. Therefore, we also study these

quantities to understand their evolution with size. These quantities are determined

within B3LYP for the lowest-energy structures obtained within the same scheme.

The vertical ionization potential (VIP) is calculated as the self-consistent energy

difference between the cluster and its positive ion with the same geometry. The VIP

is plotted in Fig. 6 As a function of cluster size. We first note the oscillating

behavior of the IP which is due to change of spin multiplicity of the ground state of

the series, and clusters with even number of electrons are closed-shell system,

whereas odd number of electrons AlPn clusters are open-shell systems. Therefore

clusters with even number of electrons present the higher values of the IP with

respect to their neighboring odd systems, because it is more difficult to remove an

electron from the doubly occupied HOMO of a closed-shell system than from a

single occupied HOMO of an open-shell system. This result is consistent with the

variation of HOMO energy along the series (See Fig. 5). In Fig. 6 we display the

VIP decreases as the cluster size increases, and the peak occurring at AlP5 is

especially prominent, with large drops for the following clusters. Also shown in

Fig. 6 are the VIPs of pure phosphorus clusters. These have also been calculated at

the B3LYP/6-311?G* level of theory, wit h structures optimized at the same level

of theory. The comparison of the two curves shows that the AlPn and Pn clusters

exhibit the same odd–even pattern. It is also interesting to note that replacing one P

in a Pn cluster with Al, to give AlPn-1, results in smaller VIPs except for AlP5

cluster.

We have calculated the adiabatic electron affinity (AEA), the vertical detachment

energy (VDE) and vertical electron affinities (VEA) of AlP2, AlP3 and AlP4 clusters

00.5

11.5

22.5

33.5

44.5

0 2 4 6 8 10 12 14

No.of P atoms

HO

MO

-LU

MO

Fig. 5 HOMO–LUMO gap inelectron volt. Solid line B3LYP.Dotted line SVWN

66.5

77.5

88.5

99.510

2 4 6 8 10 12 14

No.of atoms

Ioni

zatio

n po

tent

ial (

eV)

Fig. 6 Ionization potential inelectron volt calculated atB3LYP/6-311?G* level forAlPn and Pn clusters. Solid lineAlPn clusters. Dotted line Pn

clusters

172 L. Guo

123

(See Table 2). The result reveals that the calculated AEAs and VDEs are the closest

to the experiment given by Gomez et al. [6] in the 2001 from their anion

photoelectron spectroscopy study. Therefore, the calculation method employed is

reliable and the results are accurate enough. The calculated VEA values of AlP2,

AlP3 and AlP4 are 1.755, 1.674 and 1.581 eV, respectively. The calculated AEA,

VDE and VEA values are evaluated as the difference of total energies in

the following manner: the adiabatic electron affinity (AEA) = E(optimized

neutral) - E(optimized anion); the vertical detachment energy (VDE) = E(neutral

at optimized anion geometry) - E(optimized anion). The vertical electron affinity

(VEA) = E(optimized neutral) - E(anion at optimized neutral geometry). The

differences between VEA, AEA and VDE are due to the change in the geometry

between AlPn and AlPn- (n = 2–4) anion. We have also calculated vertical electron

affinities (VEA) for AlPn clusters (See Fig. 7). The VEA also exhibits an odd–even

pattern. This is again a consequence of the electron pairing effect. In the case of

clusters with an even number of valence electrons, the extra electron has to go into

the next orbital, which costs energy, resulting in a lower value of VEA. The VEA

curve shows a conspicuous peak at AlP8. Large drops in VEA after AlP6 and AlP10

are also evident. A comparison of the VEAs of AlPn clusters and pure phosphorus

clusters again shows the same odd–even pattern. This observation is consistent with

the observations from VIPs.

Another useful quantity is the chemical hardness [14, 26], which can be

approximated as

g � 1=2 I � Að Þ � 1=2ðeL � eHÞ; ð5Þ

Table 2 The adiabatic electron affinity (AEA), the vertical detachment energy (VDE) and the vertical

electron affinities (VEA) of AlP2, AlP3 and AlP4 clusters in eV

Cluster AEA VDE VEA

AlP2 (calculated) 1.973 2.218 1.755

AlP2 (experimental)a 1.933 ± 0.007 2.21 ± 0.025

AlP3 (calculated) 1.954 2.376 1.674

AlP3 (experimental)a 2.06 ± 0.05 2.58 ± 0.025

AlP4 (calculated) 2.623 3.431 1.581

AlP4 (experimental)a 2.64 ± 0.05 3.40 ± 0.025

a Ref. [6]

00.5

11.5

22.5

33.5

2 4 6 8 10 12 14

No.of atoms

Ele

ctro

n af

finity

(eV

)

Fig. 7 Electron affinity inelectron volt calculated atB3LYP/6-311?G* level. Solidline AlPn clusters. Dotted line Pn

clusters

Density Functional Study 173

123

where A and I are the electron affinity and ionization potential, eL and eH are the

energies of the highest occupied molecular orbital (HOMO) and the lowest unoc-

cupied molecular orbital (LUMO), respectively. Chemical hardness has been

established as an electronic quantity that in many cases may be used to characterize

the relative stability of molecules and aggregates through the principle of maximum

hardness (PMH) proposed by Pearson [25]. The PMH asserts that molecular systems

at equilibrium present the highest value of hardness [27, 28]. The hardness of AlPn

clusters, calculated according to Eq. (5) using VIP for the ionization potential and

VEA for the electron affinity, is shown in Fig. 8. Assuming that the PMH holds in

these systems, we expect the hardness to present an oscillating behavior with local

maxima at the clusters with even valence-electron clusters, as found for the VIP,

VEA, and the relative energy in Figs. 4 and 8 shows that the even valence-electron

clusters present higher values of hardness than their neighboring clusters. We

observe the even–odd oscillating feature similar to that already stressed in the VIP,

VEA, and stability criteria. Stable clusters are harder than their neighbors odd

valence-electron systems.

We present in Table 3 the static mean polarizability \a[ and mean polarizabil-

ity per atom (\a[/n ? 1) for the lowest-energy structures calculated within the

B3LYP scheme. The static mean polarizability \a[ is calculated from the

polarizability tensor components as

\a[ ¼ 1

3ðaXX þ aYY þ aZZÞ ð6Þ

The static polarizability represents one of the most important observables for the

understanding of the electronic properties of clusters, it is proportional to the

number of electrons of the systems, and it is very sensitive to the delocalization of

valence electrons as well as to the structure and shape of the system. In Table 3 we

note that when going from AlP2 to AlP12 the the polarizability of the clusters

increases monotonically showing the expected proportionality with n (or the total

electrons number). We note in Table 3 that the mean polarizability per atom (\a[/

n ? 1) of AlPn clusters decreases from 45.8 a.u. for AlP2 to 25.4 a.u. for AlP12,

with the lowest value (24.1 a.u.) for AlP7. The lowest value of polarizability per

atom occurs for AlP7, which could be due to a combined effect of the compactness

of structure and the electronic shell closure that occurs for this cluster. The closed-

shell electronic configuration of AlP7 would result in the low response of the

electrons to the applied electric field, resulting, thereby, in lower value of

polarizability. Chattaraj et al. [29–31] have proposed a minimum polarizability

22.22.42.62.8

33.23.43.63.8

0 2 4 6 8 10 12 14

No.of P atoms

Har

dnes

s (e

V)

Fig. 8 Hardness of AlPn

clusters in electron voltcalculated within the B3LYP

174 L. Guo

123

principle (MPP) which states that the natural direction of evolution of any system is

toward a state of minimum polarizability. There are many studies confirming the

validity of the MPP on different kind of reactions and systems. So we can speculate

the AlP7 cluster is a stable cluster. It is also evident from Table 3 that the odd–even

oscillations, which were present in the binding energy, dissociation energy, VIP,

VEA, and hardness, are not seen here. The mean polarizability per atom seems to

saturate beyond AlP7.

Summary and Conclusions

Phosphorus clusters doped with a single Al impurity atom have been studied by an

all-electron linear combination of atomic orbital approach, within spin-polarized

density functional theory, using both the LSDA and hybrid GGA schemes for the

exchange–correlation. A reversal of the order of isomers with LSDA and GGA

occurs for small clusters with n. The Al impurity is found to occupy a peripheral

position. The stability of the lowest-energy structures is investigated by analyzing

energies. Odd–even oscillations are observed in most of the physical properties

investigated, suggesting that clusters with an even number of electrons are more

stable than their odd-electron neighboring clusters. These odd–even effects are

especially prominent within the B3LYP scheme. The stability analysis and the

various electronic structure properties indicate the AlP5 and AlP7 clusters to be the

more stable clusters among those studied.

Acknowledgments This work was financially supported by the National Natural Science Foundation of

China (Grant No. 20603021), Youth Foundation of Shanxi (2007021009) and the Youth Academic

Leader of Shanxi.

Table 3 Static mean polarizadbility \a[ and mean polarizability per atom (\a[/n ? 1) of AlPn clus-

ters calculated within the B3LYP with (6-311?G*) basis set

System axx ayy azz \a[ \a[/n ? 1

AlP2 60.0 95.4 119.2 91.5 45.8

AlP3 68.3 153.5 119.0 113.6 28.4

AlP4 110.5 138.0 147.9 132.1 26.4

AlP5 166.3 166.3 148.0 160.2 26.7

AlP6 158.4 244.3 156.1 186.3 26.6

AlP7 186.8 170.8 220.2 192.6 24.1

AlP8 265.0 242.9 214.2 240.7 26.7

AlP9 241.7 320.2 229.7 263.9 26.4

AlP10 354.1 261.4 239.0 284.8 25.9

AlP11 462.9 259.6 247.6 323.3 26.9

AlP12 450.3 246.3 294.0 330.2 25.4

All values are in a.u.

Density Functional Study 175

123

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