Number density function nonlinear functionals

In the previous Maxwelhan description of X-ray diffraction, the electron number density n(r, t) was considered to be a known function of r,t. In reality, this density is modulated by the laser excitation and is not known a priori. However, it can be determined using methods of statistical mechanics of nonlinear optical processes, similar to those used in time-resolved optical spectroscopy [4]. The laser-generated electric field can be expressed as E(r, t) = Eoo(0 exp(/(qQr ot)), where flo is the optical frequency and q the corresponding wavevector. The calculation can be sketched as follows. 

Thus, the ratio is dependent on experimental parameters such as the optical path length, sample number density, and the phase matching conditions for the intermediate third-order processes, as well as the ratio of the third-and fifth-order response functions. The ratio of the response functions is directly related to the magnitude of the nonlinearity in the system, which is reflected by the magnitude of the potential anharmonicity, g(3), and the nonlinearity in the polarizability, a,2>. For example, let us consider only the NP contribution to the direct fifth-order response [Equation (21)]. For simplicity we will consider a system represented by a single mode, in other words the response is isotropic. If we express the third-order response functions in term of the coordinate [Equation (17)] and ignore all higher order terms,... 

The net rate of bubble generation, H, describes redistribution of mass in bubble-bubble interactions. Thus, H is a nonlinear functional of F(x,m,t) and Equations (2) and (3) are a pair of coupled, nonlinear, integro-differential equations in the bubble number density, similar to Boltzmann s equation in the kinetic theory of gases (26,27) or to Payatakes et al (22) equations of oil ganglia dynamics. 

Banerjee and Harbola [69] have worked out a variation perturbation method within the hydrodynamic approach to the time-dependent density functional theory (TDDFT) in order to evaluate the linear and nonlinear responses of alkali metal clusters. They employed the spherical jellium background model to determine the static and degenerate four-wave mixing (DFWM) y and showed that y evolves almost linearly with the number of atoms in the cluster. 

During the last 10-20 years, a large number of efficient theoretical methods for the calculation of linear and nonlinear optical properties have been developed— this development includes semi-empirical, highly correlated ab initio, and density functional theory methods. Many of these approaches will be reviewed in later chapters of this book, and applications will be given that illustrate the merits and limitations of theoretical studies of linear and nonlinear optical processes. It will become clear that theoretical studies today can provide valuable information in Are search for materials with specific nonlinear optical properties. First, there is the possibility to screen classes of materials based on cost and time effective calculations rather then labor intensive synthesis and characterization work. Second, there is Are possibility to obtain a microscopic understanding for the performance of the material—one can investigate the role of individual transition channels, dipole moments, etc., and perform systematic model Improvements by inclusion of the environment, relativistic effects, etc. 

Density Functional Theory and the Local Density Approximation Even in light of the insights afforded by the Born-Oppenheimer approximation, our problem remains hopelessly complex. The true wave function of the system may be written as i/f(ri, T2, T3,. .., Vf ), where we must bear in mind, N can be a number of Avogadrian proportions. Furthermore, if we attempt the separation of variables ansatz, what is found is that the equation for the i electron depends in a nonlinear way upon the single particle wave functions of all of the other electrons. Though there is a colorful history of attempts to cope with these difficulties, we skip forth to the major conceptual breakthrough that made possible a systematic approach to these problems. 

Since Xpo follows a distribution and the relation between A.po and the hole radius r is nonlinear [Eq. (11.3)], we prefer to estimate the mean hole-volume as the mean of the number-weighted hole volume distribution. The radius distribution [the probability density function (pdf)], n(rfc), can be calculated from n (rh) = - 3 (A.) (dx/drh) [Gregory, 1991 Deng et al., 1992b] ... 

Several useful methods have been proposed to overcome the variational coUapse problem, and a number of different schemes have been proposed for obtaining SCF wave functions for excited states [10, 16-26]. In recent years, there has been renewed interest in the orthogonality-constrained methods [14, 27] as weU as in the SCF theory for excited states [28-32]. It is clear that an experience accumulated for the HF excited state calculations can be useful to develop similar methods within density functional theory [33-36]. Some of these approaches [10, 18, 19, 23, 24, 26, 30-35] explicitly introduce orthogonality constraints to lower states. Other methods [21, 22, 25] either use this restriction implicitly or locate excited states as higher solutions of nonlinear SCF equations [29]. In latter type of scheme, the excited state SCF wave functions of interest are not necessarily orthogonal to the best SCF functions for a lower state or states of the same symmetry. 

At larger intensities I, the density of the lower state can noticeably decrease while the upper state density Nj increases. This corresponds to the strong field case in Sect.2.9.4 and means that N. (I) and N (I) are functions of I and therefore dl is no longer proportional to I [nonlinear absorption). We shall illustrate this nonlinear absorption by the simple example of a two-level system with population densities and N2 and equal statistical weights = 92 = total number density N = + N2 of the... 

The non-BO wave functions of different excited states have to differ from each other by the number of nodes along the internuclear distance, which in the case of basis (49) is r. To accurately describe the nodal structure in aU 15 states considered in our calculations, a wide range of powers, m, had to be used. While in the calculations of the H2 ground state [119], the power range was 0 0, in the present calculations it was extended to 0-250 in order to allow pseudoparticle 1 density (i.e., nuclear density) peaks to be more localized and sharp if needed. We should notice that if one aims for highly accurate results for the energy, then the wave function of each of the excited states must be obtained in a separate calculation. Thus, the optimization of nonlinear parameters is done independently for each state considered. 

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