Sodium clusters polarizability

In O Fig. 16-1, the experimental data series of sodium cluster polarizabilities are plotted... 

The temperature in the canonical BOMD simulation was controlled by a Nos6-Hoover chain thermostat (Hoover 1985 Martyna et al. 1992 Nos6 1984). In order to study the temperature dependency of the sodium cluster polarizabilities the polarizability tensor a was calculated along the recorded trajectories. For this purpose the first 20 ps of each trajectory were discarded and a was then calculated in 100 fs time steps along the remaining 200 ps. Due to the computational demand of the analytical polarizability calculation along the BOMD trajectories we employed the LDA kernel. Thus, the computational level for the calculation of the temperature dependent part of the cluster polarizabilities was VWN/TZVP-FIP/GEN-A2. The temperature dependent mean sodium cluster polarizability was then calculated as ... 

In this section, the results obtained by the first systematic study of the temperature dependency of the sodium cluster polarizabilities at a reliable first-principle all-electron level of theory were reviewed. The main results of this study are summarized as follows ... 

The calculated finite temperature sodium cluster polarizabilities show characteristic minima at the dimer and octamer as expected from the jellium model. 

Individual molecular structures besides these two are not resolved in the calculated finite temperature sodium cluster polarizabilities. 

There are exceptions from this trend. A noted example is the polarizability of sodium clusters significantly underestimated by semi-local functionals.73... 

Table 3. Electric dipole polarizabilities in units of R, of neutral sodium clusters in the spherical jellium model (SJM) and in a jellium model with finite surface thickness (FSJM)...

Finite differences, finally, is the simplest approach the total energy and dipole moment are computed as a function of the strength of the external electric field. Then, a quadratic fit to the energy or, equivalently, a linear fit to the dipole moment as a function of the external field amplitude provide the polarizability a. This direct approach has been used for sodium clusters in Refs [110] (LSD A) and [111] (LSD A and GGA), and for aluminum in Ref. [106] (LSDA). 

In particular, the pronounced oscillating behavior observed by Knight et al. up to the hex-amer (O Fig. 16-la, dots) was not confirmed by the more recent study of Rayane and co-workers (O Fig. 16-la, squares). To emphasize the spread between these two experimental data sets we have connected the data points in O Fig. 16-1 by vertical lines. This figure also shows that for the larger sodium clusters with 7,8 and 9 atoms excellent agreement between the reported data sets exist. The heptamer and octamer polarizabilities were also measured by Tikhonov et al. (2001) and are in good agreement with the depicted experimental data in O Fig. 16-1, too. 

It is well established in the literature that calculated DFT polarizabilities at this level of theory differ by no more than 5% from experiment (Calaminici et al. 1998). O Fig. 16-la, however, shows that the calculated polarizabilities of the sodium clusters are not only considerably too low but even at the level of the gradient corrected PBE functional differences of more than 10% relative to the experimental values can occur. By and large these results are confirmed by many other theoretical calculations. 

Blundell, S. A., Guet, C., Zope, R. R. (2000). Temperature dependence of the polarizability of sodium clusters. Physical Review Letters, 84, 4826. 

Calaminici, R, Koster, A. M., Gamboa Martinez, G. U. (2007b). Temperature dependence of the polarizability of sodium clusters An all-electron density functional study. In G. Maroulis T. Simos (Eds.), Computational methods in science and engineering, theory and computation Old problems and new challenges (Vol. 1, pp. 207-211). New York AIP Conference Proceedings Melville. 

Chandrakumar, K. R. S., Ghanty, T. K., Ghosh, S. K. (2004). Static dipole polarizability and binding energy of sodium clusters Na ( = 1 - 10) A critical assessment of all-electron based post-Hartree-Fock and density functional methods. Journal of Chemical Physics, 120, 6487. 

Atomic-level response of sodium clusters to external electric field has also been studied recently [55] using a decomposition of the total cluster dipole moment and polarizability into contributions from atomic volumes. The atomic dipole moments and polarizabilities thus obtained have also been partitioned into the atomic dipole and charge-transfer components. The relative contribution of these two components as a function of the size and shape of the clusters have been studied. Also the contributions are shown to depend on the location of the atomic site in the cluster. Thus, the surface atoms have been shown to have larger contribution to the polarizability than the interior ones. The anisotropy of the total polarizabilities is also shown to correlate with the shape anisotropy of the clusters. These ab initio results on the atomic charge and dipole components that contribute to the overall cluster polarizability thus would enable one to validate the coarse-grained DFT-based results in a more detailed fashion since the coarse-grained approaches provide the atomic charges and atomic dipoles besides the overall polarizability. 

Most of the electronic properties of molecules and clusters depend on their shape and size. Even though there are varied definitions for shape and size of a species, a widely used definition is the MED topological one. Boyd has used the MED contour to discuss the relative sizes of atoms [49]. On the other hand for molecules, Bader et al. [32] proposed the surface of constant electron density (0.002 a.u.) to describe shape and size of diatomic molecules. Here in the present paper, for justifying the variation in polarizability and to quantify the delocalization of the valence electrons in these clusters, the volume enclosed within the 0.0001 a.u. MED isosurface is quantified. The MED contours at 0.01,0.001, and 0.0001 a.u. of the Lij, Li4, Na2, and Na4 are depicted in Eigure 11.8. The respective volume enclosed within the 0.0001 a.u. MED isosurfaces of lithium and sodium clusters with sizes ranging from 2 through 40 is reported in Table 11.7. 

II. 8, it is clear that the volume enclosed within the isosurfaces of lithium is less than that of corresponding sodium cluster, as may be expected. The volumes enclosed within the 0.0001 a.u. MED isosurface of larger clusters reported in Table 11.7 show a trend similar to that of polarizability. Moreover, Chandrakumar et al. [33] have... 

A Comparison of Theoretical (Experimental) Polarizability, a (in a.u. ) of the Most Stable Lithium and Sodium Clusters and the Volume (in a.u. ) Enclosed within the 0.0001 a.u. MED-lsosurface at B3LYP/6-31 -i-G(d) Level of Theory... 

The electric dipole polarizability of sodium clusters has been extensively studied, both experimentally [99-101] and theoretically [102-104]. In addition, the polarizability of the sodium atom is accurately known. The latest experimental value is... 

Thakkar and Lupinetti5 have used the coupled-cluster method in conjunction with the Douglas-Kroll relativistic Hamiltonian to obtain a very accurate value for the static dipole polarizability of the sodium atom. Their revised value for a(Na) = 162.88 0.6 au resolves a previous discrepancy between theory and experiment and when combined with an essentially exact value for lithium, establishes the ratio a(Li)/a(Na) = 1.0071 0.0037, so that, because of the... 

Chandrakumar, K.R.S., Ghanty, T.K. and Ghosh, S.K (2004) Relationship between ionization potential, polarizability, and softness a case study of lithium and sodium metal clusters. J. Phys. Chcm. A, 108, 6661-6666. 

Antoine R, Rayane D, AUouche AR et al (1999) Static dipole polarizability of small mixed sodium-lithium clusters. J Chem Phys 110 5568-5577... 

A precise theoretical and experimental determination of polarizability would provide an important probe of the electronic structure of clusters, as a is very sensitive to the presence of low-energy optical excitations. Accurate experimental data for a wide range of size-selected clusters are available only for sodium, potassium [104] and aluminum [105, 106]. Theoretical predictions based on DFT and realistic models do not cover even this limited sample of experimental data. The reason for this scarcity is that the evaluation of polarizability by the sum rule (46) requires the preliminary computation of S(co), which, with the exception of Ref. [101], is available only for idealized models. Two additional routes exist to the evaluation of a, in close analogy with the computation of vibrational properties static second-order perturbation theory and finite differences [107]. Again, the first approach has been used exclusively for the spherical jellium model. In this case, the equations to be solved are very similar to those introduced in Ref. [108] for the computation of atomic polarizabilities. Applications of this formalism to simple metal clusters are reported, for instance, in Ref. [109]. 

The conventional approach to calculate the polarizability of metal clusters is to solve the Kohn-Sham equations using suitable approximate forms for the exchange correlation functionals and a finite field method. We have recently carried out a systematic all electron DFT-based calculations for the polarizability and binding energy of sodium as well as lithium metal clusters [51,52]. It has been shown that the effect of electron correlation plays a significant role in determining the polarizability of metal clusters, although the effect is less pronounced for lithium clusters. Electron... 

In this paper, we investigate the possibilities offered by widely used DFT methods. We have chosen test cases in three different, difficult classes of problems (1) the linear and nonlinear polarizabilities of metal clusters, (2) the polarizabilities of novel compounds, and (3) the interaction-induced polarizability in weakly bonded systems. In particular, the three test cases are the sodium tetramer, a particularly soft molecule, the new compound HXel, and the interaction polarizability of two water molecules in the dimer (H20)2. 

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