## Introduction

Benichou

et al. measured the static electric dipole polarizabilities of lithium clusters made of

n (

n=2-22) atoms [

1]. The experiment consisted of deflecting a collimated cluster beam through a static inhomogeneous electric field. The strong decrease per atom from Li to Li

_{3}-Li

_{4} showed that electronic delocalization was reached for very small sizes. Moreover, directly measured polarizabilities were consistent with photoabsorption data. They thus confirmed unambiguously the

missing optical strength in lithium clusters. Maroulis and Xenides reported highly accurate

ab initio calculations with specially designed basis sets for Li

_{4} [

2]. The molecule emerged as a particularly soft system, with very anisotroic dipole polarizability and very large second dipole hyperpolarizability. An extensive investigation of basis set and electron correlation effects led to values of

$\overline{\alpha}$ = 387.01 and

$\Delta \alpha $ = 354.60 a.u. The mean hyperpolarizability was

$\overline{\gamma}$ = 2394·10

^{3} a.u.. They also discussed the computational aspects of the effort in view of the extension of quantumchemical studies to larger lithium clusters. Their values for the mean dipole polarizabilitiy were systematically higher than the recently reported experimental static value (326.6 a.u.) of this important quantity [

1].

Fuentealba presented a theoretical study of the static dipole polarizability of carbon clusters C

_{n} with

n ≤ 8 [

3]. They calculated the dipole polarizabilities using density functionals of the hybrid type in combination with the finite field method. They investigated large basis sets in order to obtain reliable results. They showed that the dipole polarizabilities are an important quantity for the identification of clusters with different numbers of atoms and even for the separation of isomers. In particular, they predicted that the jet formed by the two isomers of C

_{6}, cyclic and linear, would split up in the presence of an electric field. Fuentealba and Reyes calculated the dipole polarizability of a series of clusters of the type Li

_{n}H

_{m} using density functional methods [

4]. They explained the study of the trends in the mean polarizability and the anisotropy in terms of the interplay between electronic and geometrical effects. They also discussed the changes in the polarizability for different isomers of a given cluster as well as its variations when hydrogen atoms were added to a given cluster. They also calculated a very related quantity, the hardness, in the simple approximation of hardness equal to the energy gap. They discussed their values in terms of the possible stability of the different clusters.

Jackson

et al. used a first-principles, density-functional-based method to calculate the electric polarizabilities and dipole moments for several low-energy geometries of Si clusters in the size range 10 ≤

n ≤ 20 [

5]. They found that the polarizability per atom is a slowly varying, nonmonotonic function of

n. Over this size range the polarizability appeared to be correlated most strongly to cluster shape and not with either the dipole moment or the highest occupied—lowest unoccupied molecular-orbital gap. The calculations indicated that the polarizability per atom for Si clusters approaches the bulk limit from above as a function of size. Deng

et al. calculated the polarizabilities of Si clusters with 9 to 28 atoms using a density functional cluster method [

6,

7]. They based the atomic geometries on those carefully optimized by energy optimization. The polarizability showed fairly irregular variation with cluster size, but all calculated values were higher than the polarizability of a dielectric sphere with bulk dielectric constant and equivalent volume.

Hohm

et al. deduced an experimental value of 116.7 ± 1.1 a.u. for the static dipole polarizability of As

_{4} from the analysis of refractivity measurements in arsenic vapour [

8]. This was in close agreement with the theoretical result of 119.5 ± 3.6 a.u., obtained from

ab initio finite-field many-body perturbation theory and coupled-cluster calculations.

In a previous paper, the following metal clusters and fullerenes were calculated: Sc

_{n} (1 ≤ n ≤ 7 and

n = 12), C

_{n} (

n = 1, 12, 60, 70 and 82) and endohedral

[email protected]_{60} and Sc

_{n}@C

_{82} (1 ≤ n ≤ 3) [

9]. In the present paper, the following metal clusters have been calculated: Sc; Sc

_{2} linear (D

_{∞}_{h}); Sc

_{3} triangle (D

_{3h}); Sc

_{4} in three conformations, square (D

_{4h}), rhomb (D

_{2h}) and tetrahedron (T

_{d}); Sc

_{5} triangular bipyramid (D

_{3h}); Sc

_{6} in two conformations, octahedron (O

_{h}) and antiprism (D

_{3d}); Sc

_{7} pentagonal bipyramid (D

_{5h}); Sc

_{12} icosahedron (I

_{h}); Sc

_{17} (hexagonal close packing, HCP) and Sc

_{74} (HCP). For some clusters, several isomers have been considered. Atom-atom contact distances have been held at 2.945Å. The following fullerene models have been studied: C, C

_{12} (I

_{h}), C

_{60} (I

_{h}), C

_{70} (D

_{5h}) and C

_{82} (C

_{2}). The following one-shell graphite models have been computed: C

_{n} (

n =1, 6, 10, 13, 16, 19, 22, 24, 42, 54, 84 and 96). Atom-atom contact distances have been held at the experimental distance of 1.415Å. In the next section, the description of the electrostatic properties used in this study is presented. Next, the interacting induced dipoles polarization model for the calculation of molecular polarizabilities is presented. Following that, results are presented and discussed. The last section summarizes my conclusions.

## Calculation results and discussion

The molecular dipole-dipole polarizabilities <α> for the Sc

_{n }clusters are reported in

Table 1. Program POLAR gives <α> results that are one third of the corresponding PAPID reference values. This is due to a limitation in the current parametrization of POLAR, that will be improved in a future paper. In particular, the numerical restricted Hartree-Fock (RHF) value for Sc

_{1} calculated by Stiehler and Hinze (22.317Å

^{3}) is significatively above the POLAR value but of the same order of magnitude as the PAPID one [

50].

**Table 1.**
Elementary dipole-dipole polarizabilities for clusters.

**Table 1.**
Elementary dipole-dipole polarizabilities for clusters.
Sc_{n} | <α> (Å^{3})^{a} | <α> ref.^{b} |
---|

Sc | 5.631 | 16.893 |

Sc_{2} | 1.418 | 13.744 |

Sc_{3} | 2.103 | 11.557 |

Sc_{4} D_{4h} | 2.111 | 12.873 |

Sc_{4} D_{2h} | 2.461 | 11.163 |

Sc_{4} T_{d} | 2.657 | 10.041 |

Sc_{5} | 3.116 | 9.690 |

Sc_{6} O_{h} | 3.367 | 10.330 |

Sc_{6} D_{3d} | 3.904 | 10.330 |

Sc_{7} | 3.429 | 9.321 |

Sc_{12} | 3.891 | 8.724 |

Sc_{17} h.c.p. | 3.590 | 25.278 |

Sc_{74} h.c.p. | 3.630 | 23.471 |

The variation of the computed values for the elementary polarizability of Sc

_{n} clusters with the number of atoms is illustrated in

Figure 1. The PAPID results for both Sc

_{6} isomers are nearly equal and so they are superposed. On varying the number of atoms, the clusters show numbers indicative of particularly polarizable structures. Despite the PAPID results for the small clusters tend to the bulk limit, both clusters in the HCP structure comes away from this limit. Thus, both HCP cluster results should be taken with care.

**Figure 1.**
Average atom-atom polarizabilities per atom of Sc_{n} clusters vs. cluster size. Dotted lines correspond to the bulk polarizabilities.

**Figure 1.**
Average atom-atom polarizabilities per atom of Sc_{n} clusters vs. cluster size. Dotted lines correspond to the bulk polarizabilities.

As a reference, the bulk limit for the polarizability has been included, estimated from the Clausius-Mossotti relationship:

where

v is an elementary volume per atom in the crystalline state and

$\epsilon $ is the bulk dielectric constant. For metals,

$\epsilon $ approaches infinite and the dependence of

$\alpha $ with

$\epsilon $ disappears. In this work, the

v value used for Sc is 15.0Å

^{3} per atom,

$\alpha $ is calculated as 3.581Å

^{3} per atom and for C, ν=5.3Å

^{3} and

$\alpha $ is obtained as 1.265Å

^{3}.

The polarizability trend for the Sc

_{n} clusters as a function of size is different from what one might have expected. In general, the Sc

_{n} clusters calculated with POLAR are less polarizable than what one might have inferred from the bulk polarizability. Although an exception occurs for Sc

_{1} the trend is clearer after this cluster. Previous experimental work [

51] yielded the same trend for Si

_{n}, Ga

_{n}As

_{m} and Ge

_{n}Te

_{m} somewhat larger clusters. However, the Sc

_{n} clusters computed with PAPID are more polarizable than what is inferred from the bulk,

i.e., the polarizability of clusters tend to be greater than the bulk limit and approach this limit from above. Moreover, previous theoretical work with density functional theory within the one-electron approximation yielded the same trend for Si

_{n}, Ge

_{n} and Ga

_{n}As

_{m} small clusters [

52]. At present, the origin of this difference is problematic. One might argue that smaller clusters need not behave like those of intermediate size. In addition, the error bars in the experiments are quite large.

The high polarizability of the Sc

_{n} clusters (PAPID) is attributed to dangling bonds at the surface of the cluster. Indeed, most of the atoms within clusters reside on the surface. In fact, these structures are thought more closely related to the high-pressure metallic phases than to the diamond structure [

53]. For example, it has been shown that the polarizabilities of alkali clusters significantly exceed the bulk limit and tend to decrease with increasing cluster size [

54,

55].

The geometries of the C

_{n} fullerene models have been optimized with MMID [

35,

36]. The polarizabilities <α> for the fullerene models are summarized in

Table 2.

**Table 2.**
Elementary dipole-dipole polarizabilities for fullerene models.

**Table 2.**
Elementary dipole-dipole polarizabilities for fullerene models.
C_{n} fullerene | <α> (Å^{3})^{a} | <α> ref.^{b} |
---|

C | 0.588 | 1.322 |

^{C}12 | 0.978 | 0.722 |

^{C}60 | 0.782 | 0.904 |

^{C}70 | 0.781 | 0.920 |

^{C}82 | 0.763 | 0.911 |

In general, POLAR underestimates <α>. In particular, the numerical RHF value for C

_{1} (1.783Å

^{3}) [

50] and the density functional theory (DFT) value calculated by Fuentealba (1.882Å

^{3}) [

3] are significatively above POLAR but on the same order of magnitude as PAPID. For C

_{60}, the experimental elementary value measured by Antoine

et al. (1.28±0.13Å

^{3}) [

56] and the

ab initio value calculated by Norman

et al. (1.430Å

^{3}) [

57] are somewhat greater than those obtained with POLAR and PAPID.

The fullerene models calculated with POLAR and PAPID are, in general, less polarizable than what is inferred from the bulk and approach this limit from below (see

Figure 2). Although an exception occurs for C

_{1} (PAPID) or C

_{12} (POLAR) this trend is clearer after this structure.

**Figure 2.**
Average atom-atom polarizabilities per atom of fullerene models vs. cluster size.

**Figure 2.**
Average atom-atom polarizabilities per atom of fullerene models vs. cluster size.

The polarizabilities <α> for the C

_{n} one-shell graphite models are listed in

Table 3.

**Table 3.**
Elementary dipole-dipole polarizabilities for one-shell graphite models.

**Table 3.**
Elementary dipole-dipole polarizabilities for one-shell graphite models.
C_{n} graphite | <α> (Å^{3})^{a} | <α> ref.^{b} |
---|

C | 0.588 | 1.322 |

C_{6} | 0.746 | 1.024 |

^{C}10 | 0.775 | 1.067 |

^{C}13 | 0.789 | 1.074 |

^{C}16 | 0.795 | 1.091 |

^{C}19 | 0.805 | 1.109 |

^{C}22 | 0.798 | 1.116 |

^{C}24 | 0.796 | 1.117 |

^{C}42 | 0.813 | 1.185 |

^{C}54 | 0.839 | 1.212 |

^{C}84 | 0.851 | 1.273 |

^{C}96 | 0.875 | 1.293 |

POLAR underestimates <α>. In particular, the DFT value for C

_{6}-cyclic (1.445Å

^{3}) is doubled with respect to POLAR but only somewhat greater than that obtained with PAPID [

3]. The graphite models calculated with POLAR and PAPID are, in general, less polarizable than what is inferred from the bulk and approach this limit from below (see

Figure 3). Although an exception occurs for C

_{1} (PAPID) this trend is clearer after this structure.

**Figure 3.**
Average atom-atom polarizabilities per atom of one-shell graphite models vs. cluster size.

**Figure 3.**
Average atom-atom polarizabilities per atom of one-shell graphite models vs. cluster size.

When comparing Sc

_{n} and C

_{n}, <α> is greater for the three-dimensional (3D) Sc

_{n} clusters than for the two-dimensional C

_{n} clusters. This is due to the 3D character of the

metallic bond in Sc

_{n}. The <α> is greater for the planar C

_{n} graphite models than for the curved fullerene models due to the weakening of the

π bonds in the non-planar fullerene structure (see Section Electrostatic properties). For all the clusters in

Table 1,

Table 2 and

Table 3, the mean relative error is -39%. It should be noted that this error improves to -34% if the exception structures are eliminated from the tables. In order to prevent these small figures caused by the algebraic sum, the mean unsigned relative error has been calculated. This error results 44% and decreases to 40% without the exception structures.

## Conclusions

A method for the calculation of the molecular dipole-dipole polarizability <α> has been presented and applied to Sc_{n} and C_{n} (fullerene and one-shell graphite) model clusters. From the preceding results the following conclusions can be drawn:

1. On varying the number of atoms, the clusters show numbers indicative of particularly polarizable structures. The polarizability also depends on the chosen isomer.

2. The results of the present work clearly indicate that due to the differences between POLAR and PAPID results it may become necessary to recalibrate POLAR. It appears that the results of good quality ab initio calculations might be suitable as primary standards for such a calibration. Work is in progress on the recalibration of POLAR.

3. The polarizability trend for the clusters as a function of size is different from what one might have expected. The small Sc_{n} clusters (POLAR) and the large C_{n} fullerenes (both POLAR and PAPID) are less polarizable than what one might have inferred from the bulk polarizability. The Sc_{n} clusters (PAPID) are more polarizable than what is inferred from the bulk. The high polarizability of the Sc_{n} clusters (PAPID) is attributed to arise from dangling bonds at the surface of the cluster. Recommended elementary polarizability values are 17—22Å^{3} (Sc_{n}), 1.8—1.9Å^{3} (small C_{n}-fullerene), and 1.3—1.9Å^{3} (small C_{n}-graphite) and 1.3Å^{3} (big C_{n}-fullerene and C_{n}-graphite).