Polarizabilities Of Clusters Research Articles

Theoretical study of small two-dimensional gold clusters

The geometric and electronic properties of small two-dimensional (2D) ${\mathrm{Au}}_{N}$ $(N=2\char21{}20)$ clusters are studied using the density-functional method. The lowest-energy geometries of a closed-packed nature are selected for 2D ${\mathrm{Au}}_{2}\ensuremath{-}{\mathrm{}\mathrm{Au}}_{8}$ clusters. A clear odd-even oscillation is found for the stability and energy gaps of ${\mathrm{Au}}_{N}$ clusters with $Nl~15.$ The polarizabilities of the 2D clusters are calculated and found to be strongly anisotropic. In the normal direction of a 2D cluster, the polarizability is smaller than that of bulk Si, indicating a nonmetallic character. A fitting formula is proposed to estimate the polarizability of a 2D Au cluster with arbitrary size and shape in the normal direction. The behavior of a 2D Au cluster in a strong electric field is also discussed.

Structures and stabilities of small silicon clusters: Ab initio molecular-orbital calculations of Si7–Si11

Ab initio all-electron molecular-orbital calculations have been carried out to study the structure and relative stability of small silicon clusters (Sin, n=7–11). A number of low-energy geometric isomers are optimized at the second-order Møller–Plesset (MP2) MP2/6-31G(d) level. Harmonic vibrational analysis has been performed to assure that the optimized geometries are stable. The total energies of stable isomers are computed at the coupled-cluster single and double substitutions (including triple excitations) [CCSD(T)] CCSD(T)/6-31G(d) level. The calculated binding energies per atom at both the MP2/6-31G(d) and CCSD(T)/6-31G(d) levels agree with the experiments. For Si7, Si8, and Si10, the lowest-energy structures are the same as those predicted previously from the all-electron optimization at the Hartree–Fock (HF) HF/6-31G(d) level [Raghavachari and Rohlfing, J. Chem. Phys. 89, 2219 (1988)]. For Si9, the lowest-energy isomer is same as that predicted based on density-functional plane-wave pseudopotential method [Vasiliev, Ogut, and Chelikowsky, Phys. Rev. Lett. 78, 4805 (1997)]. Particular attention has been given to Si11 because several low-energy geometric isomers were found nearly isoenergetic. On the basis of MP2/6-311G(2d)//CCSD(T)/6-311G(2d) calculation, we identified that the C2v isomer, a tricapped trigonal prism with two additional caps on side trigonal faces, is most likely the global-minimum structure. However, another competitive geometric isomer for the global minimum is also found on basis of the MP2/6-311G(2d)//CCSD(T)/6-311G(2d) calculation. Additionally, calculations of the binding energy and the cluster polarizability offer more insights into relatively strong stability of two magic-number clusters Si6 and Si10.

On the theoretical determination of the static dipole polarizability of intermediate size silicon clusters

The B3PW91 method of the density functional theory has been applied to the study of the dipole polarizability of medium size silicon clusters employing pseudopotential on all of them. All electron calculations have been performed for those clusters with less than nine atoms. In addition, we have optimized the structures of the clusters with less than ten atoms. On using the modified genetic algorithm, fourteen conformers of silicon isomers with nine atoms have been determined. The corresponding geometry of these clusters was optimized and their relative stability determined. The calculated polarizabilities are compared with experimental data and previous theoretical results.

Dipole polarizabilities of germanium clusters

Dipole polarizabilities of Ge n clusters with 2–25 atoms are calculated using finite field (FF) method within density functional theory. The dipole moments and polarizabilities of clusters are sensitively dependent on the cluster geometries and electronic structures. The clusters with low symmetry and large HOMO–LUMO gap prefer to large dipole moments. The polarizabilities of the Ge n clusters increase rapidly in the size range of 2–5 atoms and then fluctuate around the bulk value. The larger HOMO–LUMO gap may lead to smaller polarizability. As compared with the compact structure and diamond structure, the prolate cluster structure corresponds to a larger polarizability.

Equilibrium properties of clusters in the harmonic superposition approximation

The inherent structure approach, or superposition approach is used to calculate some physical and chemical properties of clusters in thermodynamic equilibrium, for both the classical and quantum regimes. In the harmonic approximation some simple analytical estimates are obtained for the average polarizability of metal clusters, and for the photoabsorption intensity of chromophore-doped van der Waals clusters. Comparison with Monte Carlo simulations reveals good agreement, which could be further improved using systematic corrections for anharmonic perturbations.

Molecular polarizability of fullerenes and endohedral metallofullerenes

AbstractThe interacting induced dipoles polarization model implemented in our program POLAR is used for the calculation of the molecular dipole µ and tensor quadrupole $\overline{\overline{{\mit \Theta}}}$ moments and also the dipole–dipole polarizability $\overline{\overline{\alpha}}$. The method is tested with Sc n, C n (fullerene and graphite) and endohedral Sc n@C m clusters. The polarizability is an important quantity for the identification of clusters with different numbers of atoms and even for the separation of isomers. The results for the polarizability are of the same order of magnitude as from reference calculations performed with our version of the program PAPID. The bulk limit for the polarizability is estimated from the Clausius–Mossotti relationship. The polarizability trend for these clusters as a function of size is different from what one might have expected. The clusters are more polarizable than what one might have inferred from the bulk polarizability. Previous theoretical work yielded the same trend for Si n, Ge n and Ga nAs m small clusters. However, previous experimental work yielded the opposite trend for Si n, Ga nAs m and Ge nTe m larger clusters. At present, the origin of this difference is problematic. One might argue that smaller clusters need not behave like those of intermediate size. The high polarizability of small clusters is attributed to dangling bonds at the surface of the cluster. In this respect, semiconductor clusters resemble metallic clusters. Copyright © 2002 John Wiley & Sons, Ltd.

Reinvestigation of Hydrogen Bond Effects on the Polarizability and Hyperpolarizability of Urea Molecular Clusters

The oriented-gas model based on the additivity hypothesis is widely used in predicting the macroscopic nonlinear optical susceptibility of a molecular crystal from molecular (hyper)polarizability calculations. Here, we argue that the intermolecular hydrogen bond interactions will break the additivity relationship for the first hyperpolarizability of urea hydrogen-bonded clusters on the basis of our high-level ab initio studies. The calculations at the HF level without electron correlation correction result in the underestimation of the hydrogen bond effect on the first hyperpolarizabilities to about 15% in the urea case. On the other hand, both HF and MP2 results show the additivity of linear polarizability of urea clusters. We hope that the study will be helpful to the areas of molecular design and simulation of novel nonlinear optical materials.

Medium perturbations on the molecular polarizability calculated within a localized dipole interaction model

We have studied the medium effects on the frequency-dependent polarizability of water by separating the total polarizability of water clusters into polarizabilities of the individual water molecules. A classical frequency-dependent dipole–dipole interaction model based on classical electrostatics and an Unsöld dispersion formula has been used. It is shown that the model reproduces the polarizabilities of small water clusters calculated with time-dependent density functional theory. A comparison between supermolecular calculations and the localized interaction model illustrate the problems arising from using supermolecular calculations to predict the medium perturbations on the solute polarizability. It is also noted that the solute polarizability is more dependent on the local geometry of the cluster than on the size of the cluster.

Electric dipole moments and polarizabilities of single excess electron sodium fluoride clusters: Experiment and theory

In this article we present the first measurement of the electric dipole susceptibility of one excess electron NanFn−1 clusters. The static electronic polarizability and the permanent electric dipole of these clusters have been calculated with a one-electron model. Experimental values for the susceptibility are clearly related to the calculated values of the permanent dipole. The size evolution of the dipole moments is interpreted in terms of the interplay between the lattice and electron properties. This study outlines that the response of the cluster to the electric field cannot be fully understood with only equilibrium structure calculations and that the coupling between the permanent dipole and the vibrational motion of the cluster has to be taken into account.

Polarizability of molecular clusters as calculated by a dipole interaction model

We have developed and investigated a dipole interaction model for calculating the polarizability of molecular clusters. The model has been parametrized from the frequency-dependent molecular polarizability as obtained from quantum chemical calculations for a series of 184 aliphatic, aromatic, and heterocyclic compounds. A damping of the interatomic interaction at short distances is introduced in such a way as to retain a traceless interaction tensor and a good description of the damping over a wide range of interatomic distances. By adopting atomic polarizabilities in addition to atom-type parameters describing the damping and the frequency dependence, respectively, the model is found to reproduce the molecular frequency-dependent polarizability tensor calculated with ab initio methods. A study of the polarizability of four dimers has been carried out: the hydrogen fluoride, methane, benzene, and urea dimers. We find in general good agreement between the model and the quantum chemical results over a wide range of intermolecular distances. To demonstrate the power of the model, the polarizability has been calculated for a linear chain of urea molecules with up to 300 molecules and one- and two-dimensional clusters of C60 with up to 25 molecules. Substantial intermolecular contributions are found for the polarizability anisotropy, whereas the effects are small for the mean polarizability. For the mean polarizability of C60, we find good agreement between the model and experiments both in the case of an isolated molecule and in a comparison of a planar cluster of 25 C60 molecules with experimental results on thin films.

Nonempirical calculation of linear and nonlinear polarizability of TeO2-based molecular clusters and piezoelectric properties of crystalline tellurium oxide

Polarization characteristics of the paratellurite TeO2 crystal were studied in the model of the molecular cluster represented by the (TeO2)3 molecular chain. Nonempirical calculations of linear and nonlinear polarizability reveal a steep growth of the second hyperpolarizability when passing on from the single TeO2 molecule to the molecular chain (TeO2)3, which may be related to intermolecular contacts. Linear polarizability of isostructural molecules SO2, SeO2, and TeO2 have an equal order of magnitude, while the second hyperpolarizability of TeO2 is somewhat higher than that of SeO2 and exceeds the SO2 polarizability by an order of magnitude.

Velocity distribution measurement and two-wire field effects for electric deflection of a neutral supersonic cluster beam

The basic principle of deflection of a beam of polarizable particles by an inhomogeneous electric field is straightforward, but its application to realistic beams having spatial and velocity distributions of finite widths requires special care. We present a detailed treatment of these issues as developed for a measurement of electric polarizabilities of alkali clusters in a continuous supersonic beam. We begin by describing a general technique for extracting the beam velocity distribution from the time-of-flight profile generated by two separate 50–50 gating choppers. This method yields accurate model-free information directly from the experimental profile, without errors associated with detector delays, and with high signal throughput. We then use a simple but accurate approximation for the deflecting field variation over space, and derive analytical expressions for the deflection of beams of finite width and finite velocity spread. This allows us to evaluate the magnitude of a number of corrections to the ideal formula; the results are applicable to optimization of various electric and magnetic field deflection experiments. Finally, we demonstrate, both theoretically and experimentally, the phenomenon of local velocity variations across the beam profile which arise as a result of deflection.

Molecular polarizability of semiconductor clusters and nanostructures

The interacting-induced-dipoles polarization model implemented in program PAPID is used for the calculation of the molecular dipole–dipole polarizability α ̄ ̄ . The method is tested with Si n, Ge n and Ga n As m small clusters. On varying the number of atoms, the clusters show numbers indicative of particularly polarizable structures. The results for the polarizability are in agreement with reference calculations from Chelikowsky. The bulk limit for the polarizability is estimated from the Clausius–Mossotti relationship. The polarizability trend for these clusters as a function of size is different from what one might have expected. The clusters are all more polarizable than what one might have inferred from the bulk polarizability. Previous experimental work has yielded the opposite trend for somewhat larger clusters, i.e., in this work the polarizability of clusters tends to be lower than the bulk limit and approaches this limit from below. At present, the origin of this difference is problematic. One might argue that smaller clusters need not behave like those of intermediate size. The high polarizability of small clusters is attributed to dangling bonds at the surface of the cluster. Indeed, most of the atoms within small clusters reside on the surface. In this respect, semiconductor clusters resemble metallic clusters.

Size, Order, and Dimensional Relations for Silicon Cluster Polarizabilities

Response theory calculations in the random phase approximation are applied to linear polarizabilities and second hyperpolarizabilities of 1-, 2-, and 3-dimensional hydrogen-terminated silicon clusters. Successive enlargement of the clusters to embody on the order of 50 silicon atoms plus bond-saturating hydrogen atoms allows for extrapolation to bulk values of individual silicon atom contributions in the 1D and 3D cases. Modern effective core potentials are shown to provide excellent approximations to the all-electron values in all cases; errors for both polarizabilities and hyperpolarizabilities are on the order of 1%. The findings indicate considerable time savings in predictions of the electric polarizability properties of elements beyond the first row atoms.

A measurement of the polarizability of sodium clusters

The static electric dipole polarizabilities of sodium clusters ${\mathrm{Na}}_{n}$ for a series of sizes in the range $7<~n<~93$ were measured by electric field deflection of a supersonic cluster beam. Results for the smaller clusters are in overall good agreement with previous experiments. Polarizabilities of the larger clusters continue the downward trend with increasing size, but remain considerably above the bulk limit. There is a degree of correlation between the polarizability values and electronic shell structure. Furthermore, there is an indication that the polarizability is sensitive to the source conditions, possibly reflecting the influence of internal cluster temperature.

Electric dipole polarizabilities of Ni12–58

The electric dipole polarizabilities of Ni12–Ni58 have been measured via a molecular beam deflection experiment. The majority of the clusters display polarizabilities α between 4 and 6 Å3 per atom, below the atomic value of 6.8 Å3 and above that predicted for ideal conducting spheres. Local extrema in α/atom are observed for certain Nin. In particular, those clusters that adopt closely packed or quasispherical structures (e.g., Ni38, Ni55) display local minima, while Ni21, Ni22, Ni25, and Ni49–54, each an icosahedron or polyicosahedron with “missing” atoms, display anomalously high values. The relationship between the polarizabilities of nickel clusters and their shapes and structures is discussed.

Ab Initio Study of the Structure and Polarizability of Sulfur Clusters, Sn (n = 2−12)

The structure and the dipole polarizabilities of Sn clusters (n = 2−12) have been calculated using density functional theory within the B3LYP approximation and conventional ab initio Hartree−Fock (HF) and coupled-cluster with single and double and perturbative triple excitations (CCSD(T)) methods. The results show that the binding energy per atom increases with the size of the cluster and reaches the asymptotic limit for a relatively small n value. There is an excellent agreement between B3LYP and CCSD(T) data in predicting the energy of the disproportionation reaction 2Sn → Sn-1 + Sn+1, which indicates that S2, S6, and S8 are especially stable, in agreement with the experiment. 〈α〉 increases with n and linearly correlates with the molecular volume. 〈α〉/n increases with n and reaches the asymptotic limit per n → ∞ from below, contrary to what happens in small semiconductor and metallic clusters. A well-defined correlation between 〈α〉 and hardness is not found, while the 〈αn〉 − n〈α1〉 difference value linearly correlates with the atomization energy. In the sulfur clusters, the minimum polarizability principle does not hold, the lone-pair electron polarizability being more diffuse, hence more polarizable, in the cluster than in the free atom. Pure vibrational effects on 〈α〉 are negligible.

Ab initio structures and polarizabilities of sodium clusters

We present quantitative ab initio calculations for Na cluster structures and polarizabilities, for all cluster sizes up to 20 atoms. Our calculations are performed by combining an ab initio core-corrected pseudopotential and a gradient-corrected density functional within a real space approach. We find the cluster bonding to be very floppy and catalog a host of low-energy quasi-degenerate isomers for all second-decade clusters. The existence of these isomers results in a band of polarizability values for each cluster size even at zero temperature. This eliminates any finer structure in the polarizability curve. We further show that the experimental polarizability values are consistently underestimated by calculations at zero temperature. By computing the effects of structure expansion and distortion due to a finite temperature we arrive at a quantitative agreement between theory and experiment.

Molecular Polarizability of Sc and C (Fullerene and Graphite) Clusters

A method (POLAR) for the calculation of the molecular polarizability <α> is presented. It uses the interacting induced dipoles polarization model. As an example, the method is applied to Scn and Cn (fullerene and one-shell graphite) model clusters. On varying the number of atoms, the clusters show numbers indicative of particularly polarizable structures. The <α> are compared with reference calculations (PAPID). In general, the Scn calculated (POLAR) and Cn computed (POLAR and PAPID) are less polarizable than what is inferred from the bulk. However, the Scn calculated (PAPID) are more polarizable than what is inferred. Moreover, previous theoretical work yielded the same trend for Sin, Gen and GanAsm small clusters. The high polarizability of the Scn clusters (PAPID) is attributed to arise from dangling bonds at the surface of the cluster.

Influence of the exchange-correlation potential in density-functional calculations on polarizabilities and absorption spectra of alkali-metal clusters

All-electron calculations in large basis sets of excitation energies, oscillator strengths, and polarizabilities of small alkali-metal clusters of Li, Na, and K, with up to eight atoms, are performed using time-dependent density-functional theory. It is shown that the use of the recently developed statistical average of orbital potentials (SAOP) exchange-correlation (xc) potential [P.R.T. Schipper et al., J. Chem. Phys. 112, 1344 (2000)] leads to polarizabilities of these alkali-metal clusters which are 10\char21{}15 % larger than polarizabilities calculated with the xc potential of the local-density approximation (LDA). The lower LDA polarizabilities (in comparison to the SAOP) are shown to originate from differences in the low-lying excitation energies, which are determined by the xc potential in the molecular inner and valence region. In spite of such differences, both SAOP and LDA results are shown to provide reliable assignments of the experimental absorption spectra, with typical errors in peak positions of only 0.1\char21{} 0.2 eV, or even less.