Many-electron systems correlation densities

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. 

Density Functional Models. Methods in which the energy is evaluated as a function of the Electron Density. Electron Correlation is taken into account explicitly by incorporating into the Hamiltonian terms which derive from exact solutions of idealized many-electron systems. 

The simplest way to gain a better appreciation for tlie hole function is to consider the case of a one-electron system. Obviously, the Lh.s. of Eq. (8.6) must be zero in that case. However, just as obviously, the first term on the r.h.s. of Eq. (8.6) is not zero, since p must be greater than or equal to zero throughout space. In die one-electron case, it should be clear that h is simply the negative of the density, but in die many-electron case, the exact form of the hole function can rarely be established. Besides die self-interaction error, hole functions in many-electron systems account for exchange and correlation energy as well. 

Recently there has been a great deal of interest in nonlinear phenomena, both from a fundamental point of view, and for the development of new nonlinear optical and optoelectronic devices. Even in the optical case, the nonlinearity is usually engendered by a solid or molecular medium whose properties are typically determined by nonlinear response of an interacting many-electron system. To be able to predict these response properties we need an efficient description of exchange and correlation phenomena in many-electron systems which are not necessarily near to equilibrium. The objective of this chapter is to develop the basic formalism of time-dependent nonlinear response within density functional theory, i.e., the calculation of the higher-order terms of the functional Taylor expansion Eq. (143). In the following this will be done explicitly for the second- and third-order terms... 

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

Krieger, Chen, lafrate, and Savin (KCIS) [172] revisited the LDA for correlation and attributed the LDA overestimation of correlation energies for nonuniform densities to the fact that, unlike an electron gas, finite many-electron systems have a nonzero energy gap between the Fermi level and the continuum (the gap is equal to the ionization potential [173]). To improve the LDA description of systems with a finite orbital gap, they made use of the formula of Rey and Savin [174] for the correlation energy per particle of the electron gas with an energy gap G,... 

We begin with the most important issue. It is assumed that in principle it is possible to obtain a full description of many-electron systems by DFT if the exact (yet unknown) density-dependent exchange-correlation functional is employed. This seemingly undisputed tenet, based on the Hohenberg-Kohn theorem, was recently called into question [106, 107]. In his works Kaplan shows that the conventional Kohn-Sham equations are invariant with the respect to the total spin,... 

At that time, the best-known method for electron correlation in molecules was undoubtedly configuration interaction (Cl). This tool had developed in the hands of Slater and Condon, with early applications by Boys, Parr, Matsen, and their coworkers around 1950 (see Ref. [2] for an excellent review). Somewhat less known to the quantum chemistry community was the parallel development in the mid 1950s of the correlation problem in physics that originated with Brueckner [3] and Goldstone [4], termed many-body perturbation theory (MBPT) because it was applicable to many-electron systems. This feature, that we now call size-extensivity [5], was not shared by Cl, but was a necessity for the physics applications to nuclear matter and the electron gas. Important questions at this time included the correlation treatment of the high- and low-density electron... 

The density functional (DF) method of electronic structure calculations is based on the feet that the ground state of a many-electron system is completely determined by its electron density. When compared with correlated ab initio methods, the density functional method can be applied to much larger systems than those approachable by traditional ab initio methods. The density functional method has long been employed in studying electronic structures of solid-state systems and has become a widely applicable, advanced computational method in chemistry. In this section, we briefly comment on the concepts of DF theory and compare them with those of HF theory. 

The approximation inherent in the Hartree-Fock theory is the lack of proper inclusion of electronic correlation. But the equations are solvable even for many-electron systems. Today the most popular way of treating such systems approximately is through the density functional theory, where the correlation is accounted for by more or less empirical means (see Chapter 10). 

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