Dielectric function theories

The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external held to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser held, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach. 

Bergman effective medium theory, in conjunction with a model dielectric function for the particles, has been used. For the other layers the Bruggeman effective medium theory was used. After [Bel6]. 

Figure 6 Singly occupied molecular orbital (SOMO) of a propeller-like trimer radical anion of acetonitrile obtained using density functional theory. The structure was immersed in a polarizable dielectric continuum with the properties of liquid acetonitrile. Several surfaces (on the right) and midplane cuts (on the left) are shown. The SOMO has a diffuse halo that envelops the whole cluster within this halo, there is a more compact kernel that has nodes at the cavity center and on the molecules.

The quantum-mechanical expression for the dielectric function may be written in the form (see, e.g., Ziman, 1972, Chap. 8, for an elementary discussion of the quantum theory of optical properties)... 

I he notation 0e indicates that this is the dielectric function at frequencies low i ompared with electronic excitation frequencies. We have also replaced co0 with l (, the frequency of the transverse optical mode in an ionic crystal microscopic theory shows that only this type of traveling wave will be readily excited bv a photon. Note that co2 in (9.20) corresponds to 01 e2/me0 for the lattice vibrations (ionic oscillators) rather than for the electrons. The mass of an electron is some thousands of times less than that of an ion thus, the plasma liequency for lattice vibrations is correspondingly reduced compared with that lor electrons. 

The reflectance, dielectric functions, and refractive indices, together with calculations based on the Drude theory, for the common metal aluminum are shown in Fig. 9.11. Aluminum is described well by the Drude theory except for the weak structure near 1.5 eV, which is caused by bound electrons. The parameters we have chosen to fit the reflectance data, hu>p = 15 eV and hy = 0.6 eV, are appreciably different from those used by Ehrenreich et al. (1963), hup = 12.7 eV and hy = 0.13 eV, to fit the low-energy (hu < 0.2 eV) reflectance of aluminum. This is probably caused by the effects of band transitions and the difference in electron scattering mechanisms at higher energies. The parameters we use reflect our interest in applying the Drude theory in the neighborhood of the plasma frequency. 

The free-electron contribution to the dielectric function in Fig. 9A2b is obtained from the Drude theory with parameters determined from the low-... 

There is no oscillation the polarization merely relaxes toward zero with a time constant t. In the following paragraphs, we shall use (9.35), the basic assumption of the Debye theory, to derive an expression for the dielectric function of a collection of permanent dipoles. 

The Debye equations (9.42) are particularly important in interpreting the large dielectric functions of polar liquids one example is water, the most common liquid on our planet. In Fig. 9.15 measured values of the dielectric functions of water at microwave frequencies are compared with the Debye theory. The parameters tod, e0v, and r were chosen to give the best fit to the experimental data r = 0.8 X 10 -11 sec follows immediately from the frequency at which e" is a maximum e0d — e0v is 2e"ax. 

The dielectric function of a metal can be decomposed into a free-electron term and an interband, or bound-electron term, as was done for silver in Fig. 9.12. This separation of terms is important in the mean free path limitation because only the free-electron term is modified. For metals such as gold and copper there is a large interband contribution near the Frohlich mode frequency, but for metals such as silver and aluminum the free-electron term dominates. A good discussion of the mean free path limitation has been given by Kreibig (1974), who applied his results to interpreting absorption by small silver particles. The basic idea is simple the damping constant in the Drude theory, which is the inverse of the collision time for conduction electrons, is increased because of additional collisions with the boundary of the particle. Under the assumption that the electrons are diffusely reflected at the boundary, y can be written... 

---