Polarizabilities dynamic frequency-dependent

In linear, spherical and synnnetric tops the components of a along and perpendicular to the principal axis of synnnetry are often denoted by a and respectively. In such cases, the anisotropy is simply Aa = tty -If the applied field is oscillating at a frequency w, then the dipole polarizability is frequency dependent as well a(co). The zero frequency limit of the dynamic polarizability a(oi) is the static polarizability described above. 

The core element of LR-TDDFT is the relation between the dynamic (frequency dependent) polarizability of the system under investigation and the quantities derivable from Kohn-Sham equations. LR-TDDFT has been subject to many reviews (for fundamental aspects see the recent review by Gross and Marquard112, for the original description of the adaptation of LR-TDDFT to molecular systems see the review by Casida111). Before discussing the response of an embedded electron density, the key elements of LR-TDDFT will be provided here1. 

Choosing a non-zero value for uj corresponds to a time-dependent field with a frequency u, i.e. the ((r r)) propagator determines the frequency-dependent polarizability corresponding to an electric field described by the perturbation operator QW = r cos (cut). Propagator methods are therefore well suited for calculating dynamical properties, and by suitable choices for the P and Q operators, a whole variety of properties may be calculated. " ... 

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. 

The extension of density functional theory (DFT) to the dynamical description of atomic and molecular systems offers an efficient theoretical and computational tool for chemistry and molecular spectroscopy, namely, time-dependent DFT (TDDFT) [7-11]. This tool allows us to simulate the time evolution of electronic systems, so that changes in molecular structure and bonding over time due to applied time-dependent fields can be investigated. Its response variant TDDF(R)T is used to calculate frequency-dependent molecular response properties, such as polarizabilities and hyperpolarizabilities [12-17]. Furthermore, TDDFRT overcomes the well-known difficulties in applying DFT to excited states [18], in the sense that the most important characteristics of excited states, the excitation energies and oscillator strengths, are calculated with TDDFRT [17, 19-26]. 

The experimental measures of these molecular electric properties involve oscillating fields. Thus, the frequency-dependence effects should be considered when comparing the experimental results . Currently, there are fewer calculations of the frequency-dependent polarizabilities and hyperpolarizabilities than those of the static properties. Recent advances have enabled one to study the frequency dispersion effects of polyatomic molecules by ab initio methods In particular, the frequency-dependent polarizability a and hyperpolarizability y of short polyenes have been computed by using the time-dependent coupled perturbed Hartree-Fock method. The results obtained show that the dispersion of a increases with the increase in the optical frequency. At a given frequency, a and its relative dispersion increase with the chain length. Also, like a, the hyperpolarizability y values increase with the chain length. While the electronic static polarizability is smaller than the dynamic one, the vibrational contribution is smaller at optical frequencies. ... 

F1F and DFT methods have been used by Smith et al.205 to calculate the dipole polarizabilities, a, of the all trans polyenes up to CigH2o, and polyacenes to CigH12 and their molecular ions. The static values and the dynamic response at 800 nm have been calculated. For the smaller molecules the results have been compared with those of correlated methods. It is found that the uncorrelated results are about 20% higher than those of the correlated methods for the neutral molecules but are very similar for the molecular ions. General conclusions about the scaling and frequency dependence of a values obtained by simpler methods are drawn. 

The situation is somewhat different for the convergence with the wavefunction model, i.e. the treatment of electron correlation. As an anisotropic and nonlinear property the first dipole hyperpolarizability is considerably more sensitive to the correlation treatment than linear dipole polarizabilities. Uncorrelated methods like HF-SCF or CCS yield for /3 results which are for small molecules at most qualitatively correct. Also CC2 is for higher-order properties not accurate enough to allow for detailed quantitative studies. Thus the CCSD model is the lowest level which provides a consistent and accurate treatment of dynamic electron correlation effects for frequency-dependent properties. With the CC3 model which also includes the effects of connected triples the electronic structure problem for j8 seems to be solved with an accuracy that surpasses that of the latest experiments (vide infra). 

For the pyrimidine molecule we also calculated the frequency dependence of the Sj state polarizability up to the first resonance. The change of the dynamic polarizability upon excitation is displayed in Fig. 1, for the a, a, and components, respectively. The dispersion of the ground state polarizabilities in the same frequency interval as for the excited state does not differ significantly from... 

Here and Ss are the frequency-dependent dielectric constants of the metal and the solution, respectively. Using known pyridine polarizability and silver dielectric data, large enhancements could be obtained (up to 10 ). In terms of the molecular picture, a several-eV decrease of level spacing was involved. This shift, however, strongly depends on the frequency, through the dielectric constant of the metal. This is a dynamic shift and the resonance is really a joint metal-molecule-photon excitation. This is different from a shift of levels under static fields. This point has often been misunderstood. 

It is apparent that non-linear-optical processes rely on a dynamic or frequency-dependent property. I will therefore, in general, restrict this article to calculations made at this level and, with one or two exceptions, I will consider only ab initio theory. Much work has been done on the static hyper-polarizabilities (as well as the static and dynamic polarizability a) but in order to make a direct connection with experiment, I have chosen to exclude this work. Also, again with one or two exceptions, I will only deal with molecules and exclude atoms. 

The response to frequency-dependent external fields may be obtained from Hartree-Fock response theory, yielding dynamical polarizabilities and hyperpolarizabilities. The identification of excitation energies as the poles of the dynamical polarizability tensor may be invoked to calculate excitation energies as well as one-photon and two-photon transition moments from the time development of the ground state [40-42]. 

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. 

The theoretical result obtained for a continuous medium sphere confirms the computational results Eloc(x) x/R and is linearly proportional to the applied field Eo. The dispersion of values for Eioc(x) around the x/R line in Fig. 1 is due to the presence of Stone-Wales defects that induce some dispersion of the direction of Eloc which is perpendicular to the surface on the average. Furthermore, the fullerenes studied here are far from being a continuous surface due to their small size, which explains the deviation with the continuous model. The perspective of this approach is to extend these calculations with a frequency-dependent model by including dynamical polarizabilities and kinetic energy for dipoles and charges. 

In another work, Stener et solved Eqs. (95) and (96) self-consistently for some closed-shell atoms. From the frequency-dependent changes in the electron density they calculated the dynamic polarizability... 

An interesting alternative approach is the direct calculation by polarization propagator, or linear response, methods.The poles of the propagator yield the transition frequencies, the residues yield the corresponding transition moments, and the propagator itself determines linear response properties such as the frequency-dependent (or dynamic) polarizability. 

This derivation does not hold for a field that oscillates rapidly with time. In such a field, the induced dipole oscillates and the amplitude of these oscillations depends on the frequency. We can, however, define a frequency-dependent dynamic polarizability, or molecular electric... 

Eshuis et al have implemented fully propagated time-dependent Hartree-Fock theory to calculate the real time electronic dynamics of closed- and open-shell molecules in strong oscillating electric fields. This method has been illustrated on the determination of the frequency-dependent polarizability of ethylene and is shown to converge, in the weak field limit, to the same results as the linearized TDHF method. 

Polarizability is a general concept that quantifies the response of an electron cloud of an ion to the apphcation of a time-dependent electromagnetic field resulting in a frequency-dependent polarizability. Our strict concern is with static, or zero-frequency polarizability as variations of an electric field induced by thermal fluctuations of an electrolyte operate at timescales much larger than the timescales of inner dynamics of an electron cloud. Frequency-dependent polarizability leads to other interesting effects, such as the London forces [32], when spontaneous fluctuations of electronic structure of two molecules become correlated at close spacial separations. These interactions, however, play secondary role when compared to induced interactions that arise from static polarizability [33, 34]. 

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