Atomic static polarizability

Kang, Y.K. and Jhon, M.S. (1982). Additivity of Atomic Static Polarizabilities and Dispersion Coefficients. Theor.Chim.Acta, 61,41-48. 

Let us have a look at the prediction of the mean molecular polarizability a , , by neural networks and RDF descriptors. can be calculated from additive contributions of the atomic static polarizability a of individual atoms i... 

TOO Schwerdtfeger, P. (2006) Atomic Static Dipole Polarizabilities, in Computational Aspects of Electric Polarizability Calculations Atoms, Molecules and Clusters (ed. G. Maroulis), Imperial College Press, London, pp. 1-32. 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

This is a well-known result that has been obtained in several different ways [8, 30, 28, 31, 32] The exact correlation potential of DFT is known [28] to fall off as —a/(2r4) for atoms with spherical N and (N — l)-electron ground states, with a being the static polarizability of the (N — l)-electron ground state. Theorem 2 provides a simple way of checking how the OEP correlation-only potential VC(T(r) falls off for a given approximate orbital functional approx[ 0. ] One only needs to determine the asymptotic decay of ucjv [Pg.43]

Advances in Chemical Physics as have Bishop[18] and Luo et al.[ 19] for Advances in Quantum Chemistry. Two reviews which cover both theoretical and experimental developments have been written by Bonin and Kadar-Kallen[20] for polarizabilities and by Shelton and Rice[21] for hyperpolarizabilities. Finally, though it is not a review as such, attention is drawn to the paper by Stiehler and Hinze[22] on the calculation of static polarizabilities and hyperpolarizabilities for the atoms He through Kr. This paper is so thorough in its reference to other calculations on these atoms that it deserves to be specifically singled out as an excellent source for theoretical data. 

TABLE III. Calculated free and screened static polarizabilities for Ceo in atomic units, using different exchange-correlation potentials [97]. 

London s eqn. (15) for the dipole-dipole dispersion energy is not a simple product of properties of the separate atoms. A partial separation was achieved in 1948 by Casimir and Polder who expressed the /r dispersion energy as the product of the polarizability of each molecule at the imaginary frequency iu integrated over u from zero to infinity. The polarizability at imaginary frequencies may be a bizarre property but it is a mathematically well behaved function that decreases monotonically from the static polarizability at m = 0 to zero as u—> oo. 

Here the superscripts (BC) and (LJ) refer to the Buckingham-Corner and Lennard-Jones potentials, respectively is the distance between the m-th atom of the molecule and the s-th atom of the surface. The coefficients C, D, B, Q depend on microscopic characteristics, such as static polarizability, ionization potential, etc. For a detailed discussion of atom-atom interaction potentials, the reader is referred to [12]. The subscripts M and S denote atomic species of the adsorbed molecule and the adsorbent surface note that in summations like (11), it is implied that for any value of m (or s) there is the definite M (or S) value which corresponds to a particular atomic species. Therefore, the internal summation in equation (11) can be performed to give the sum of atomic contributions ... 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

The polarizability of an atom or molecule describes the response of the electron cloud to an external field. The atomic or molecular energy shift KW due to an external electric field E is proportional to i for external fields that are weak compared to the internal electric fields between the nucleus and electron cloud. The electric dipole polarizability a is the constant of proportionality defined by KW = -0(i /2. The induced electric dipole moment is aE. Hyperpolarizabilities, coefficients of higher powers of , are less often required. Technically, the polarizability is a tensor quantity but for spherically symmetric charge distributions reduces to a single number. In any case, an average polarizability is usually adequate in calculations. Frequency-dependent or dynamic polarizabilities are needed for electric fields that vary in time, except for frequencies that are much lower than electron orbital frequencies, where static polarizabilities suffice. 

For an interaction potential (r) = -C r are characteristic energies of the atom and surface = 1 for a free-electron metal andg = (e - l)/( + 1) for an ionic crystal Here,/ is the oscillator strength from the ground state to an excited state k, with excitation energy This formula is often used to estimate static polarizabilities (v = 0)... 

A contraction of the cluster volume with respect to that of an equivalent piece of bulk metal has also been predicted [96]. The calculated cluster radius is smaller than the radius assumed in the spherical jellium model, where the volume is the same as that of an equivalent piece cut out of a macroscopic metal. This global contraction seems to be a general feature of small metallic clusters, and is well documented experimentally [97]. The volume contraction explains the discrepancies between experimentally determined static polarizabilities of small aluminium clusters and those obtained from jellium calculations [98]. The measured polarizabilities of Aljv clusters with N < 40 are smaller than those predicted by a SJM calculation. The classical static polarizability (per atom) for... 

Another way to include polarization into force fields is via a classical Drude oscillator ( charge on a spring or COS [156]). Here, an additional particle with an associated charge is attached at the nucleus of each atom. The charge on the particle is added to the static charge of the atom. The polarizability is then determined via Eq. (3.30c),... 

Table 2. Static polarizabilities in atomic units for the first few isoelectronic series...

In general, the electric field is an oscillating ones, but when is constant it correlates with the induced dipole through the static polarizability. Moreover, for atomic systems, assuming the spherical coordinates with the radius r, a given direction of the field E will produces the perturbation potential along the azimuth direction ... 

By comparison between relations of s it follows the atomic static dipole polarizability to be (Garza Robles, 1993) ... 

Reis et al have carried out DFT calculations on the static polarizabilities and hyperpolarizabilities of bare boron clusters incuding up to 10 boron atoms. They find that the y-hyperpolarizabdity saturates when the cluster size reaches approximately five atoms. A maximum in the hyperpolarizabilty per atom occurs for the cluster containing six atoms. 

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